PART II.--KINETICS

§ 12. _Rectilinear Motion._--Let x denote the distance OP of a moving point P at time t from a fixed origin O on the line of motion, this distance being reckoned positive or negative according as it lies to one side or the other of O. At time t + [delta]t let the point be at Q, and let OQ = x + [delta]x. The _mean velocity_ of the point in the interval [delta]t is [delta]x/[delta]t. The limiting value of this when [delta]t is infinitely small, viz. dx/dt, is adopted as the definition of the _velocity_ at the instant t. Again, let u be the velocity at time t, u + [delta]u that at time t + [delta]t. The mean rate of increase of velocity, or the _mean acceleration_, in the interval [delta]t is then [delta]u/[delta]t. The limiting value of this when [delta]t is infinitely small, viz., du/dt, is adopted as the definition of the _acceleration_ at the instant t. Since u = dx/dt, the acceleration is also denoted by d²x/dt². It is often convenient to use the "fluxional" notation for differential coefficients with respect to time; thus the velocity may be represented by [.x] and the acceleration by [.u] or [:x]. There is another formula for the acceleration, in which u is regarded as a function of the position; thus du/dt = (du/dx)(dx/dt) = u(du/dx). The relation between x and t in any particular case may be illustrated by means of a curve constructed with t as abscissa and x as ordinate. This is called the _curve of positions_ or _space-time curve_; its gradient represents the velocity. Such curves are often traced mechanically in acoustical and other experiments. A, curve with t as abscissa and u as ordinate is called the _curve of velocities_ or _velocity-time curve_. Its gradient represents the acceleration, and the area ([int]udt) included between any two ordinates represents the space described in the interval between the corresponding instants (see fig. 62).

So far nothing has been said about the measurement of time. From the purely kinematic point of view, the t of our formulae may be any continuous independent variable, suggested (it may be) by some physical process. But from the dynamical standpoint it is obvious that equations which represent the facts correctly on one system of time-measurement might become seriously defective on another. It is found that for almost all purposes a system of measurement based ultimately on the earth's rotation is perfectly adequate. It is only when we come to consider such delicate questions as the influence of tidal friction that other standards become necessary.

The most important conception in kinetics is that of "inertia." It is a matter of ordinary observation that different bodies acted on by the same force, or what is judged to be the same force, undergo different changes of velocity in equal times. In our ideal representation of natural phenomena this is allowed for by endowing each material particle with a suitable _mass_ or _inertia-coefficient_ m. The product _mu_ of the mass into the velocity is called the _momentum_ or (in Newton's phrase) the _quantity of motion_. On the Newtonian system the motion of a particle entirely uninfluenced by other bodies, when referred to a suitable base, would be rectilinear, with constant velocity. If the velocity changes, this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change of momentum which it produces, we have the equation

d --- (mu) = X. (1) dt

From this point of view the equation is a mere truism, its real importance resting on the fact that by attributing suitable values to the masses m, and by making simple assumptions as to the value of X in each case, we are able to frame adequate representations of whole classes of phenomena as they actually occur. The question remains, of course, as to how far the measurement of force here implied is practically consistent with the gravitational method usually adopted in statics; this will be referred to presently.

The practical unit or standard of mass must, from the nature of the case, be the mass of some particular body, e.g. the imperial pound, or the kilogramme. In the "C.G.S." system a subdivision of the latter, viz. the gramme, is adopted, and is associated with the centimetre as the unit of length, and the mean solar second as the unit of time. The unit of force implied in (1) is that which produces unit momentum in unit time. On the C.G.S. system it is that force which acting on one gramme for one second produces a velocity of one centimetre per second; this unit is known as the _dyne_. Units of this kind are called _absolute_ on account of their fundamental and invariable character as contrasted with gravitational units, which (as we shall see presently) vary somewhat with the locality at which the measurements are supposed to be made.

If we integrate the equation (1) with respect to t between the limits t, t´ we obtain _ / t´ mu´- mu = | X dt. (2) _/ t

The time-integral on the right hand is called the _impulse_ of the force on the interval t´ - t. The statement that the increase of momentum is equal to the impulse is (it maybe remarked) equivalent to Newton's own formulation of his Second Law. The form (1) is deduced from it by putting t´- t = [delta]t, and taking [delta]t to be infinitely small. In problems of impact we have to deal with cases of practically instantaneous impulse, where a very great and rapidly varying force produces an appreciable change of momentum in an exceedingly minute interval of time.

In the case of a constant force, the acceleration [.u] or [:x] is, according to (1), constant, and we have

d²x --- = [alpha], (3) dt²

say, the general solution of which is

x = ½[alpha]t² + At + B. (4)

The "arbitrary constants" A, B enable us to represent the circumstances of any particular case; thus if the velocity [.x] and the position x be given for any one value of t, we have two conditions to determine A, B. The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line. We may take it as an experimental result, although the best evidence is indirect, that a particle falling freely under gravity experiences a constant acceleration which at the same place is the same for all bodies. This acceleration is denoted by g; its value at Greenwich is about 981 centimetre-second units, or 32.2 feet per second. It increases somewhat with the latitude, the extreme variation from the equator to the pole being about ½%. We infer that on our reckoning the force of gravity on a mass m is to be measured by mg, the momentum produced per second when this force acts alone. Since this is proportional to the mass, the relative masses to be attributed to various bodies can be determined practically by means of the balance. We learn also that on account of the variation of g with the locality a gravitational system of force-measurement is inapplicable when more than a moderate degree of accuracy is desired.

We take next the case of a particle attracted towards a fixed point O in the line of motion with a force varying as the distance from that point. If [mu] be the acceleration at unit distance, the equation of motion becomes

d²x --- = -[mu]x, (5) dt²

the solution of which may be written in either of the forms

x = A cos [sigma]t + B sin [sigma]t, x = a cos ([sigma]t + [epsilon]), (6)

where [sigma]= [root][mu], and the two constants A, B or a, [epsilon] are arbitrary. The particle oscillates between the two positions x = ±a, and the same point is passed through in the same direction with the same velocity at equal intervals of time 2[pi]/[sigma]. The type of motion represented by (6) is of fundamental importance in the theory of vibrations (§ 23); it is called a _simple-harmonic_ or (shortly) a _simple_ vibration. If we imagine a point Q to describe a circle of radius a with the angular velocity [sigma], its orthogonal projection P on a fixed diameter AA´ will execute a vibration of this character. The angle [sigma]t + [epsilon] (or AOQ) is called the _phase_; the arbitrary elements a, [epsilon] are called the _amplitude_ and _epoch_ (or initial phase), respectively. In the case of very rapid vibrations it is usual to specify, not the _period_ (2[pi]/[sigma]), but its reciprocal the _frequency_, i.e. the number of complete vibrations per unit time. Fig. 62 shows the curves of position and velocity; they both have the form of the "curve of sines." The numbers correspond to an amplitude of 10 centimetres and a period of two seconds.

The vertical oscillations of a weight which hangs from a fixed point by a spiral spring come under this case. If M be the mass, and x the vertical displacement from the position of equilibrium, the equation of motion is of the form

d²x M --- = - Kx, (7) dt²

provided the inertia of the spring itself be neglected. This becomes identical with (5) if we put [mu] = K/M; and the period is therefore 2[pi][root](M/K), the same for all amplitudes. The period is increased by an increase of the mass M, and diminished by an increase in the stiffness (K) of the spring. If c be the statical increase of length which is produced by the gravity of the mass M, we have Kc = Mg, and the period is 2[pi][root](c/g).

The small oscillations of a simple pendulum in a vertical plane also come under equation (5). According to the principles of § 13, the horizontal motion of the bob is affected only by the horizontal component of the force acting upon it. If the inclination of the string to the vertical does not exceed a few degrees, the vertical displacement of the particle is of the second order, so that the vertical acceleration may be neglected, and the tension of the string may be equated to the gravity mg of the particle. Hence if l be the length of the string, and x the horizontal displacement of the bob from the equilibrium position, the horizontal component of gravity is mgx/l, whence

d²x gx --- = - ---, (8) dt² l

The motion is therefore simple-harmonic, of period [tau] = 2[pi][root](l/g). This indicates an experimental method of determining g with considerable accuracy, using the formula g = 4[pi]²l/[tau]².

In the case of a repulsive force varying as the distance from the origin, the equation of motion is of the type

d²x --- = [mu]x, (9) dt²

the solution of which is

x = A e^(nt) + B e^(-nt), (10)

where n = [root][mu]. Unless the initial conditions be adjusted so as to make A = 0 exactly, x will ultimately increase indefinitely with t. The position x = 0 is one of equilibrium, but it is unstable. This applies to the inverted pendulum, with [mu] = g/l, but the equation (9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighbourhood of the position of unstable equilibrium.

In acoustics we meet with the case where a body is urged towards a fixed point by a force varying as the distance, and is also acted upon by an "extraneous" or "disturbing" force which is a given function of the time. The most important case is where this function is simple-harmonic, so that the equation (5) is replaced by

d²x --- + [mu]x = f cos ([sigma]1t + [alpha]), (11) dt²

where [sigma]1 is prescribed. A particular solution is

f x = ---------------- cos ([sigma]1t + [alpha]). (12) [mu] - [sigma]1²

This represents a _forced oscillation_ whose period 2[pi]/[sigma]1, coincides with that of the disturbing force; and the phase agrees with that of the force, or is opposed to it, according as [sigma]1² < or > [mu]; i.e. according as the imposed period is greater or less than the natural period 2[pi]/[root][mu]. The solution fails when the two periods agree exactly; the formula (12) is then replaced by

ft x = ---------- sin ([sigma]1t + [alpha]), (13) 2 [sigma]1

which represents a vibration of continually increasing amplitude. Since the equation (12) is in practice generally only an approximation (as in the case of the pendulum), this solution can only be accepted as a representation of the initial stages of the forced oscillation. To obtain the complete solution of (11) we must of course superpose the free vibration (6) with its arbitrary constants in order to obtain a complete representation of the most general motion consequent on arbitrary initial conditions.

A simple mechanical illustration is afforded by the pendulum. If the point of suspension have an imposed simple vibration [xi] = a cos [sigma]t in a horizontal line, the equation of small motion of the bob is

x - [xi] m[:x] = -mg --------, l

or

gx [xi] [:x] + --- = ----. (14) l l

This is the same as if the point of suspension were fixed, and a horizontal disturbing force mg[xi]/l were to act on the bob. The difference of phase of the forced vibration in the two cases is illustrated and explained in the annexed fig. 63, where the pendulum virtually oscillates about C as a fixed point of suspension. This illustration was given by T. Young in connexion with the kinetic theory of the tides, where the same point arises.

We may notice also the case of an attractive force varying inversely as the square of the distance from the origin. If [mu] be the acceleration at unit distance, we have

du [mu] u --- = - ---- (15) dx x²

whence

2[mu] u² = ----- + C. (16) x

In the case of a particle falling directly towards the earth from rest at a very great distance we have C = 0 and, by Newton's Law of Gravitation, [mu]/a² = g, where a is the earth's radius. The deviation of the earth's figure from sphericity, and the variation of g with latitude, are here ignored. We find that the velocity with which the particle would arrive at the earth's surface (x = a) is [root](2ga). If we take as rough values a = 21 × 10^6 feet, g = 32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second. If the particles start from rest at a finite distance c, we have in (16), C = - 2[mu]/c, and therefore

dx / / 2[mu](c - x) \ -- = u = - / ( ------------- ), (17) dt \/ \ cx /

the minus sign indicating motion towards the origin. If we put x = c cos² ½[phi], we find

c^(3/2) t = ------------- ([phi] + sin [phi]), (18) [root](8[mu])

no additive constant being necessary if t be reckoned from the instant of starting, when [phi] = 0. The time t of reaching the origin ([phi] = [pi]) is

[pi] c^(3/2) t1 = -------------. (19) [root](8[mu])

This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz. 2[pi]c^(3/2)/[root][mu](§ 14). We learn that if the orbital motion of a planet, or a satellite, were arrested, the body would fall into the sun, or into its primary, in the fraction 0.1768 of its actual periodic time. Thus the moon would reach the earth in about five days. It may be noticed that if the scales of x and t be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.

In any case of rectilinear motion, if we integrate both sides of the equation

du mu -- = X, (20) dx

which is equivalent to (1), with respect to x between the limits x0, x1, we obtain _ / x1 ½ mu1² - ½ mu0² = | X dx. (21) _/ x0

We recognize the right-hand member as the _work_ done by the force X on the particle as the latter moves from the position x0 to the position x1. If we construct a curve with x as abscissa and X as ordinate, this work is represented, as in J. Watt's "indicator-diagram," by the area cut off by the ordinates x = x0, x = x1. The product ½mu² is called the _kinetic energy_ of the particle, and the equation (21) is therefore equivalent to the statement that the increment of the kinetic energy is equal to the work done on the particle. If the force X be always the same in the same position, the particle may be regarded as moving in a certain invariable "field of force." The work which would have to be supplied by other forces, extraneous to the field, in order to bring the particle from rest in some standard position P0 to rest in any assigned position P, will depend only on the position of P; it is called the _statical_ or _potential energy_ of the particle with respect to the field, in the position P. Denoting this by V, we have [delta]V - X[delta]x = 0, whence

dV X = - --, (22) dx

The equation (21) may now be written

½ mu1² + V1 = ½ mu0² + V0, (23)

which asserts that when no extraneous forces act the sum of the kinetic and potential energies is constant. Thus in the case of a weight hanging by a spiral spring the work required to increase the length by x is V = [int 0 to x] Kxdx = ½Kx², whence ½Mu² + ½Kx² = const., as is easily verified from preceding results. It is easily seen that the effect of extraneous forces will be to increase the sum of the kinetic and potential energies by an amount equal to the work done by them. If this amount be negative the sum in question is diminished by a corresponding amount. It appears then that this sum is a measure of the total capacity for doing work against extraneous resistances which the particle possesses in virtue of its motion and its position; this is in fact the origin of the term "energy." The product mv² had been called by G. W. Leibnitz the "vis viva"; the name "energy" was substituted by T. Young; finally the name "actual energy" was appropriated to the expression ½mv² by W. J. M. Rankine.

The laws which regulate the resistance of a medium such as air to the motion of bodies through it are only imperfectly known. We may briefly notice the case of resistance varying as the square of the velocity, which is mathematically simple. If the positive direction of x be downwards, the equation of motion of a falling particle will be of the form

du -- = g - ku²; (24) dt

this shows that the velocity u will send asymptotically to a certain limit V (called the _terminal velocity_) such that kV² = g. The solution is

gt V² gt u = V tanh ---, x = --- log cosh ---, (25) V g V

if the particle start from rest in the position x = 0 at the instant t = 0. In the case of a particle projected vertically upwards we have

du -- = -g - ku², (26) dt

the positive direction being now upwards. This leads to

u u0 gt V² V² + u0² tan^-1 --- = tan^-1 --- - ---, x = --- log --------, (27) V V V 2g V² + u²

where u0 is the velocity of projection. The particle comes to rest when

V u0 V² / u0² \ t = --- tan^-1 ---, x = --- log ( 1 + --- ). (28) g V 2g \ V² /

For small velocities the resistance of the air is more nearly proportional to the first power of the velocity. The effect of forces of this type on small vibratory motions may be investigated as follows. The equation (5) when modified by the introduction of a frictional term becomes

[:x] = -[mu]x - k [.x]. (29)

If k² < 4[mu] the solution is

x = a e^{-t/[tau]} cos ([sigma]t + [epsilon]), (30)

where

[tau] = 2/k, [sigma] = [root]([mu] - ¼k²), (31)

and the constants a, [epsilon] are arbitrary. This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law e^(-t/[tau]). The constant [tau] is called the _modulus of decay_ of the oscillations; if it is large compared with 2[pi]/[sigma] the effect of friction on the period is of the second order of small quantities and may in general be ignored. We have seen that a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G. Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector. When k² > 4[mu], the solution of (29) is, in real form,

x = a1 e^(-t/[tau]1) + a2 e^(-t/[tau]2), (32)

where

1/[tau]1, 1/[tau]2 = ½k ± [root](¼k² - [mu]). (33)

The body now passes once (at most) through its equilibrium position, and the vibration is therefore styled _aperiodic_.

To find the forced oscillation due to a periodic force we have

[:x] + k[.x] + [mu]x = f cos ([sigma]1t + [epsilon]). (34)

The solution is

f x = --- cos ([sigma]1t + [epsilon] - [epsilon]1), (35) R

provided k[sigma]1 R = {([mu] - [sigma]1²)² + k²[sigma]1²}^½, tan[epsilon]1 = ----------------. (36) [mu] - [sigma]1²

Hence the phase of the vibration lags behind that of the force by the amount [epsilon]1, which lies between 0 and ½[pi] or between ½[pi] and [pi], according as [sigma]1² <> [mu]. If the friction be comparatively slight the amplitude is greatest when the imposed period coincides with the free period, being then equal to f/k[sigma]1, and therefore very great compared with that due to a slowly varying force of the same average intensity. We have here, in principle, the explanation of the phenomenon of "resonance" in acoustics. The abnormal amplitude is greater, and is restricted to a narrower range of frequency, the smaller the friction. For a complete solution of (34) we must of course superpose the free vibration (30); but owing to the factor e^(-t/[tau]) the influence of the initial conditions gradually disappears.

For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short. The whole effect is summed up in the value of the instantaneous impulse, which is the time-integral of the force. Thus if an instantaneous impulse [xi] changes the velocity of a mass m from u to u´ we have

mu´- mu = [xi]. (37)

The effect of ordinary finite forces during the infinitely short duration of this impulse is of course ignored.

We may apply this to the theory of impact. If two masses m1, m2 moving in the same straight line impinge, with the result that the velocities are changed from u1, u2, to u1´, u2´, then, since the impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.

m1u1´ + m2u2´ = m1u1 + m2u2. (38)

The complete determination of the result of a collision under given circumstances is not a matter of abstract dynamics alone, but requires some auxiliary assumption. If we assume that there is no loss of apparent kinetic energy we have also

m1u1² + m2u2´² = m1u1² + m2u2². (39)

Hence, and from (38),

u2´ - u1´ = -(u2 - u1), (40)

i.e. the relative velocity of the two bodies is reversed in direction, but unaltered in magnitude. This appears to be the case very approximately with steel or glass balls; generally, however, there is some appreciable loss of apparent energy; this is accounted for by vibrations produced in the balls and imperfect elasticity of the materials. The usual empirical assumption is that

u2´ - u1´ = -e(u2 - u1), (41)

where e is a proper fraction which is constant for the same two bodies. It follows from the formula § 15 (10) for the internal kinetic energy of a system of particles that as a result of the impact this energy is diminished by the amount

m1m2 ½(1 - e²) ------- (u1 - u2)². (42) m1 + m2

The further theoretical discussion of the subject belongs to ELASTICITY.

This is perhaps the most suitable place for a few remarks on the theory of "dimensions." (See also UNITS, DIMENSIONS OF.) In any absolute system of dynamical measurement the fundamental units are those of mass, length and time; we may denote them by the symbols M, L, T, respectively. They may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the gramme, centimetre and second. All other units are derived from these. Thus the unit of velocity is that of a point describing the unit of length in the unit of time; it may be denoted by LT^-1, this symbol indicating that the magnitude of the unit in question varies directly as the unit of length and inversely as the unit of time. The unit of acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly denoted by LT^-2. The unit of momentum is MLT^-1; the unit force generates unit momentum in unit time and is therefore denoted by MLT^-2. The unit of work on the same principles is ML²T^-2, and it is to be noticed that this is identical with the unit of kinetic energy. Some of these derivative units have special names assigned to them; thus on the C.G.S. system the unit of force is called the _dyne_, and the unit of work or energy the _erg_. The number which expresses a physical quantity of any particular kind will of course vary inversely as the magnitude of the corresponding unit. In any general dynamical equation the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios. This principle is often useful as a check on the accuracy of an equation.

The theory of dimensions often enables us to forecast, to some extent, the manner in which the magnitudes involved in any particular problem will enter into the result. Thus, assuming that the period of a small oscillation of a given pendulum at a given place is a definite quantity, we see that it must vary as [root](l/g). For it can only depend on the mass m of the bob, the length l of the string, and the value of g at the place in question; and the above expression is the only combination of these symbols whose dimensions are those of a time, simply. Again, the time of falling from a distance a into a given centre of force varying inversely as the square of the distance will depend only on a and on the constant [mu] of equation (15). The dimensions of [mu]/x² are those of an acceleration; hence the dimensions of [mu] are L³T^-2. Assuming that the time in question varies as a^x[mu]^y, whose dimensions are L^(x + 3y)T^(-2y), we must have x + 3y = 0, -2y = 1, so that the time of falling will vary as a^(3/2)/[root][mu], in agreement with (19).

The argument appears in a more demonstrative form in the theory of "similar" systems, or (more precisely) of the similar motion of similar systems. Thus, considering the equations

d²x [mu] d²x´ [mu]´ --- = - ----, ---- = - -----, (43) dt² x² dt´² x´²

which refer to two particles falling independently into two distinct centres of force, it is obvious that it is possible to have x in a constant ratio to x´, and t in a constant ratio to t´, provided that

x x´ [mu] [mu]´ --- : --- = ---- : -----, (44) t² t´² x² x´²

and that there is a suitable correspondence between the initial conditions. The relation (44) is equivalent to

x^(3/2) x´^(3/2) t : t´ = ------- : --------, (45) [mu]^½ [mu]´^½

where x, x´ are any two corresponding distances; e.g. they may be the initial distances, both particles being supposed to start from rest. The consideration of dimensions was introduced by J. B. Fourier (1822) in connexion with the conduction of heat.

§ 13. _General Motion of a Particle._--Let P, Q be the positions of a moving point at times t, t + [delta]t respectively. A vector [->OU] drawn parallel to PQ, of length proportional to PQ/[delta]t on any convenient scale, will represent the _mean velocity_ in the interval [delta]t, i.e. a point moving with a constant velocity having the magnitude and direction indicated by this vector would experience the same resultant displacement [->PQ] in the same time. As [delta]t is indefinitely diminished, the vector [->OU] will tend to a definite limit [->OV]; this is adopted as the definition of the _velocity_ of the moving point at the instant t. Obviously [->OV] is parallel to the tangent to the path at P, and its magnitude is ds/dt, where s is the arc. If we project [->OV] on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the _component velocities_ parallel to the axes. If x, y, z be the co-ordinates of P it is easily proved that

dx dy dz u = --, v = --, w = --. (1) dt dt dt

The momentum of a particle is the vector obtained by multiplying the velocity by the mass m. The _impulse_ of a force in any infinitely small interval of time [delta]t is the product of the force into [delta]t; it is to be regarded as a vector. The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements [delta]t into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.

Newton's Second Law asserts that change of momentum is equal to the impulse; this is a statement as to equality of vectors and so implies identity of direction as well as of magnitude. If X, Y, Z are the components of force, then considering the changes in an infinitely short time [delta]t we have, by projection on the co-ordinate axes, [delta](mu) = X[delta]t, and so on, or

du dv dw m -- = X, m -- = Y, m -- = Z. (2) dt dt dt

For example, the path of a particle projected anyhow under gravity will obviously be confined to the vertical plane through the initial direction of motion. Taking this as the plane xy, with the axis of x drawn horizontally, and that of y vertically upwards, we have X = 0, Y = -mg; so that

d²x d²y --- = 0, --- = -g. (3) dt² dt²

The solution is

x = At + B, y = -½ gt² + Ct + D. (4)

If the initial values of x, y, [.x], [.y] are given, we have four conditions to determine the four arbitrary constants A, B, C, D. Thus if the particle start at time t = 0 from the origin, with the component velocities u0, v0, we have

x = u0t, y = v0t - ½ gt². (5)

Eliminating t we have the equation of the path, viz.

v0 gx² y = --- x - ---. (6) u0 2u²

This is a parabola with vertical axis, of latus-rectum 2u0²/g. The range on a horizontal plane through O is got by putting y = 0, viz. it is 2u0v0/g. we denote the resultant velocity at any instant by [.s] we have

[.s]² = [.x]² + [.y]² = [.s]0² - 2gy. (7)

Another important example is that of a particle subject to an acceleration which is directed always towards a fixed point O and is proportional to the distance from O. The motion will evidently be in one plane, which we take as the plane z = 0. If [mu] be the acceleration at unit distance, the component accelerations parallel to axes of x and y through O as origin will be -[mu]x, -[mu]y, whence

d²x d²y --- = -[mu]x, --- = - [mu]y. (8) dt² dt²

The solution is

x = A cos nt + B sin nt, y = C cos nt + D sin nt, (9)

where n = [root][mu]. If P be the initial position of the particle, we may conveniently take OP as axis of x, and draw Oy parallel to the direction of motion at P. If OP = a, and [.s]0 be the velocity at P, we have, initially, x = a, y = 0, [.x] = 0, [.y] = [.s]0 whence

x = a cos nt, y = b sin nt, (10)

if b = [.s]0/n. The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 2[pi]/[root][mu]; moreover, the velocity at any point P is equal to [root][mu]·OD, where OD is the semi-diameter conjugate to OP. This type of motion is called _elliptic harmonic_. If the co-ordinate axes are the principal axes of the ellipse, the angle nt in (10) is identical with the "excentric angle." The motion of the bob of a "spherical pendulum," i.e. a simple pendulum whose oscillations are not confined to one vertical plane, is of this character, provided the extreme inclination of the string to the vertical be small. The acceleration is towards the vertical through the point of suspension, and is equal to gr/l, approximately, if r denote distance from this vertical. Hence the path is approximately an ellipse, and the period is 2[pi] [root](l/g).

The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox, Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of small motion then are

d²x x d²y y --- = -g ------, --- = -g ------, (11) dt² [rho]1 dt² [rho]2

where [rho]1, [rho]2, are the principal radii of curvature at O. The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 2[pi] [root]([rho]1/g), 2[pi] [root]([rho]2/g). The circumstances are realized in "Blackburn's pendulum," which consists of a weight P hanging from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of the string meets AB, we have [rho]1 = CP, [rho]2 = EP. Many contrivances for actually drawing the resulting curves have been devised.

It is sometimes convenient to resolve the accelerations in directions having a more intrinsic relation to the path. Thus, in a plane path, let P, Q be two consecutive positions, corresponding to the times t, t + [delta]t; and let the normals at P, Q meet in C, making an angle [delta][psi]. Let v (= [.s]) be the velocity at P, v + [delta]v that at Q. In the time [delta]t the velocity parallel to the tangent at P changes from v to v + [delta]v, ultimately, and the tangential acceleration at P is therefore dv/dt or [:s]. Again, the velocity parallel to the normal at P changes from 0 to v[delta][psi], ultimately, so that the normal acceleration is v d[psi]/dt. Since

dv dv ds dv d[psi] d[psi] ds v² -- = -- -- = v --, v ------ = v ------ -- = -----, (12) dt ds dt ds dt ds dt [rho]

where [rho] is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v²/[rho], respectively. Take, for example, the case of a particle moving on a smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve. If the axes of x and y be drawn horizontal and vertical (upwards), and if [psi] be the inclination of the tangent to the horizontal, we have

dv dy mv² mv -- = - mg sin [psi] = - mg --, ----- = - mg cos [psi] + R. (13) ds ds [rho]

The former equation gives

v² = C - 2gy, (14)

and the latter then determines R.

In the case of the pendulum the tension of the string takes the place of the pressure of the curve. If l be the length of the string, [psi] its inclination to the downward vertical, we have [delta]s = l[delta][psi], so that v = ld[psi]/dt. The tangential resolution then gives

d²[psi] l ------- = - g sin [psi]. (15) dt²

If we multiply by 2d[psi]/dt and integrate, we obtain

/ d[psi]\² 2g ( ------ ) = --- cos [psi] + const., (16) \ dt / l

which is seen to be equivalent to (14). If the pendulum oscillate between the limits [psi] = ±[alpha], we have

/[delta][psi]\² 2g 4g ( ------------ ) = --- (cos [psi] - cos [alpha]) = --- (sin² ½[alpha] - sin² ½[psi]); (17) \ dt / l l

and, putting sin ½[psi] = sin ½[alpha]. sin [phi], we find for the period ([tau]) of a complete oscillation

_½[pi] _½[pi] / dt / l / d[phi] [tau] = 4 | ------ d[phi] = 4 / --- · | ------------------------------------ _/0 d[phi] \/ g _/0 [root](1 - sin² ½[alpha]·sin² [phi])

/ l = 4 / ---·F1(sin ½[alpha]), (18) \/ g

in the notation of elliptic integrals. The function F1 (sin [beta]) was tabulated by A. M. Legendre for values of [beta] ranging from 0° to 90°. The following table gives the period, for various amplitudes [alpha], in terms of that of oscillation in an infinitely small arc [viz. 2[pi] [root](l/g)] as unit.

+--------------+----------++--------------+----------+ | [alpha]/[pi] | [tau] || [alpha]/[pi] | [tau] | +--------------+----------++--------------+----------+ | .1 | 1.0062 || .6 | 1.2817 | | .2 | 1.0253 || .7 | 1.4283 | | .3 | 1.0585 || .8 | 1.6551 | | .4 | 1.1087 || .9 | 2.0724 | | .5 | 1.1804 || 1.0 | [oo] | +--------------+----------++--------------+----------+

The value of [tau] can also be obtained as an infinite series, by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thus

/ l / 1² 1²·3² \ [tau] = 2[pi] / --- · ( 1 + --- sin² ½[alpha] + ----- sin^4 ½[alpha] + ... ). (19) \/ g \ 2² 2²·4² /

If [alpha] be small, an approximation (usually sufficient) is

[tau] = 2[pi] [root](l/g)·(1 + (1/16)[alpha]²).

In the extreme case of [alpha] = [pi], the equation (17) is immediately integrable; thus the time from the lowest position is

t = [root](l/g)·log tan (¼[pi] + ¼[psi]). (20)

This becomes infinite for [psi] = [pi], showing that the pendulum only tends asymptotically to the highest position.

The variation of period with amplitude was at one time a hindrance to the accurate performance of pendulum clocks, since the errors produced are cumulative. It was therefore sought to replace the circular pendulum by some other contrivance free from this defect. The equation of motion of a particle in any smooth path is

d²s --- = -g sin [psi], (21) dt²

where [psi] is the inclination of the tangent to the horizontal. If sin [psi] were accurately and not merely approximately proportional to the arc s, say

s = k sin [psi], (22)

the equation (21) would assume the same form as § 12 (5). The motion along the arc would then be accurately simple-harmonic, and the period 2[pi][root](k/g) would be the same for all amplitudes. Now equation (22) is the intrinsic equation of a cycloid; viz. the curve is that traced by a point on the circumference of a circle of radius ¼k which rolls on the under side of a horizontal straight line. Since the evolute of a cycloid is an equal cycloid the object is attained by means of two metal cheeks, having the form of the evolute near the cusp, on which the string wraps itself alternately as the pendulum swings. The device has long been abandoned, the difficulty being met in other ways, but the problem, originally investigated by C. Huygens, is important in the history of mathematics.

The component accelerations of a point describing a tortuous curve, in the directions of the tangent, the principal normal, and the binormal, respectively, are found as follows. If [->OV], [->OV´] be vectors representing the velocities at two consecutive points P, P´ of the path, the plane VOV´ is ultimately parallel to the osculating plane of the path at P; the resultant acceleration is therefore in the osculating plane. Also, the projections of [->VV´] on OV and on a perpendicular to OV in the plane VOV´ are [delta]v and v[delta][epsilon], where [delta][epsilon] is the angle between the directions of the tangents at P, P´. Since [delta][epsilon] = [delta]s/[rho], where [delta]s = PP´ = v[delta]t and [rho] is the radius of principal curvature at P, the component accelerations along the tangent and principal normal are dv/dt and vd[epsilon]/dt, respectively, or vdv/ds and v²/[rho]. For example, if a particle moves on a smooth surface, under no forces except the reaction of the surface, v is constant, and the principal normal to the path will coincide with the normal to the surface. Hence the path is a "geodesic" on the surface.

If we resolve along the tangent to the path (whether plane or tortuous), the equation of motion of a particle may be written

dv mv -- = [T], (23) ds

where [T] is the tangential component of the force. Integrating with respect to s we find _ / s1 ½ mv1² - ½ mv0² = | [T] ds; (24) _/ s0

i.e. the increase of kinetic energy between any two positions is equal to the work done by the forces. The result follows also from the Cartesian equations (2); viz. we have

m([.x][:x] + [.y][:y] + [.z][:z]) = X[.x] + Y[.y] + Z[.z], (25)

whence, on integration with respect to t,

_ / ½m([.x]² + [.y]² + [.z]²) = |(X[.x] + Y[.y] + Z[.z]) dt + const. _/ _ / = |(X dx + Y dy + Z dz) + const. (26) _/

If the axes be rectangular, this has the same interpretation as (24).

Suppose now that we have a constant field of force; i.e. the force acting on the particle is always the same at the same place. The work which must be done by forces extraneous to the field in order to bring the particle from rest in some standard position A to rest in any other position P will not necessarily be the same for all paths between A and P. If it is different for different paths, then by bringing the particle from A to P by one path, and back again from P to A by another, we might secure a gain of work, and the process could be repeated indefinitely. If the work required is the same for all paths between A and P, and therefore zero for a closed circuit, the field is said to be _conservative_. In this case the work required to bring the particle from rest at A to rest at P is called the _potential energy_ of the particle in the position P; we denote it by V. If PP´ be a linear element [delta]s drawn in any direction from P, and S be the force due to the field, resolved in the direction PP´, we have [delta]V = -S[delta]s or

[dP]V S = -----. (27) [dP]s

In particular, by taking PP´ parallel to each of the (rectangular) co-ordinate axes in succession, we find

[dP]V [dP]V [dP]V X = -----, Y = -----, Z = -----. (28) [dP]x [dP]y [dP]z

The equation (24) or (26) now gives

½ mv1² + V1 = ½ mv0² + V0; (29)

i.e. the sum of the kinetic and potential energies is constant when no work is done by extraneous forces. For example, if the field be that due to gravity we have V = fmgdy = mgy + const., if the axis of y be drawn vertically upwards; hence

½ mv² + mgy = const. (30)

This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14). Again, in the case of a force Kr towards O, where r denotes distance from O we have V = [int] Kr dr = ½Kr² + const., whence

½ mv² + ½ Kr² = const. (31)

It has been seen that the orbit is in this case an ellipse; also that if we put [mu] = K/m the velocity at any point P is v = [root][mu]·OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with the known property of the ellipse that OP² + OD² is constant.

The forms assumed by the dynamical equations when the axes of reference are themselves in motion will be considered in § 21. At present we take only the case where the rectangular axes Ox, Oy rotate in their own plane, with angular velocity [omega] about Oz, which is fixed. In the interval [delta]t the projections of the line joining the origin to any point (x, y, z) on the directions of the co-ordinate axes at time t are changed from x, y, z to (x + [delta]x) cos [omega][delta]t - (y + [delta]y) sin [omega][delta]t, (x + [delta]x) sin [omega][delta]t + (y + [delta]y) cos [omega][delta]t, z respectively. Hence the component velocities parallel to the instantaneous positions of the co-ordinate axes at time t are

u = [.x] - [omega]y, v = [.y] + [omega]z, [omega] = [.z]. (32)

In the same way we find that the component accelerations are

[.u] - [omega]v, [.v] + [omega]u, [.omega]. (33)

Hence if [omega] be constant the equations of motion take the forms

m([:x] - 2[omega][.y] - [omega]²[.x]) = X, m([:y] + 2[omega][.x] - [omega]²y) = Y, m[:z] = Z. (34)

These become identical with the equations of motion relative to fixed axes provided we introduce a fictitious force m[omega]²r acting outwards from the axis of z, where r = [root](x² + y²), and a second fictitious force 2m[omega]v at right angles to the path, where v is the component of the relative velocity parallel to the plane xy. The former force is called by French writers the _force centrifuge ordinaire_, and the latter the _force centrifuge composée_, or _force de Coriolis_. As an application of (34) we may take the case of a symmetrical Blackburn's pendulum hanging from a horizontal bar which is made to rotate about a vertical axis half-way between the points of attachment of the upper string. The equations of small motion are then of the type

[:x] - 2[omega][.y] - [omega]²x = -p²x, [:y] + 2[omega][.x] - [omega]²y = -q²y. (35)

This is satisfied by

[:x] = A cos ([sigma]t + [epsilon]), y = B sin ([sigma]t + [epsilon]), (36)

provided

([sigma]² + [omega]² - p²)A + 2[sigma][omega]B = 0, \ (37) 2[sigma][omega]A + ([sigma]² + [omega]² - q²)B = 0. /

Eliminating the ratio A : B we have

([sigma]² + [omega]² - p²)([sigma]² + [omega]² - q²) - 4[sigma]²[omega]² = 0. (38)

It is easily proved that the roots of this quadratic in [sigma]² are always real, and that they are moreover both positive unless [omega]² lies between p² and q². The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, [epsilon]. Since the equations (35) are linear, these two solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is unstable. This occurs only when the period (2[pi]/[omega]) of revolution of the arm lies between the two periods (2[pi]/p, 2[pi]/q) of oscillation when the arm is fixed.

§ 14. _Central Forces. Hodograph._--The motion of a particle subject to a force which passes always through a fixed point O is necessarily in a plane orbit. For its investigation we require two equations; these may be obtained in a variety of forms.

Since the impulse of the force in any element of time [delta]t has zero moment about O, the same will be true of the additional momentum generated. Hence the moment of the momentum (considered as a localized vector) about O will be constant. In symbols, if v be the velocity and p the perpendicular from O to the tangent to the path,

pv = h, (1)

where h is a constant. If [delta]s be an element of the path, p[delta]s is twice the area enclosed by [delta]s and the radii drawn to its extremities from O. Hence if [delta]A be this area, we have [delta]A = ½ p[delta]s = ½ h[delta]t, or

dA -- = ½h. (2) dt

Hence equal areas are swept over by the radius vector in equal times.

If P be the acceleration towards O, we have

dv dr v -- = -P --, (3) ds ds

since dr/ds is the cosine of the angle between the directions of r and [delta]s. We will suppose that P is a function of r only; then integrating (3) we find _ / ½ v² = - | P dr + const., (4) _/

which is recognized as the equation of energy. Combining this with (1) we have _ h² / -- = C - 2 | P dr, (5) p² _/

which completely determines the path except as to its orientation with respect to O.

If the law of attraction be that of the inverse square of the distance, we have P = [mu]/r², and

h² 2[mu] -- = C + -----. (6) p² [tau]

Now in a conic whose focus is at O we have

l 2 1 --- = -- ± ---, (7) p² r a

where l is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken according as the conic is an ellipse or hyperbola. In the intermediate case of the parabola we have a = [oo] and the last term disappears. The equations (6) and (7) are identified by putting

l = h²/[mu], a = ± [mu]/C. (8)

Since

h² / 2 1 \ v² = -- = [mu]( --- ± --- ), (9) p² \ r a /

it appears that the orbit is an ellipse, parabola or hyperbola, according as v² is less than, equal to, or greater than 2[mu]/r. Now it appears from (6) that 2[mu]/r is the square of the velocity which would be acquired by a particle falling from rest at infinity to the distance r. Hence the character of the orbit depends on whether the velocity at any point is less than, equal to, or greater than the _velocity from infinity_, as it is called. In an elliptic orbit the area [pi]ab is swept over in the time

[pi]ab 2[pi]a^(3/2) r = ------ = ------------, (10) ½h [root][mu]

since h = [mu]^½ l^½ = [mu]^½ ba^-½ by (8).

The converse problem, to determine the law of force under which a given orbit can be described about a given pole, is solved by differentiating (5) with respect to r; thus

h² dp P = -----. (11) p³ dr

In the case of an ellipse described about the centre as pole we have

a²b² ---- = a² + b² - r²; (12) p²

hence P = [mu]r, if [mu] = h²/a²b². This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted. But since an ellipse can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab (= h/[root][mu]) we infer that the orbit will be elliptic whatever the initial circumstances. Also the period is 2[pi]ab/h = 2[pi]/[root][mu], as previously found.

Again, in the equiangular spiral we have p = r sin[alpha], and therefore P = [mu]/r³, if [mu] = h²/sin²[alpha]. But since an equiangular spiral having a given pole is completely determined by a given point and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h = [root][mu]·sin[alpha]. Similarly, in the case of a circle with the pole on the circumference we have p² = r²/2a, P = [mu]/r^5, if [mu] = 8h²a²; but this orbit is not a general one for the law of the inverse fifth power.

In astronomical and other investigations relating to central forces it is often convenient to use polar co-ordinates with the centre of force as pole. Let P, Q be the positions of a moving point at times t, t + [delta]t, and write OP = r, OQ = r + [delta]r, [angle]POQ = [delta][theta], O being any fixed origin. If u, v be the component velocities at P along and perpendicular to OP (in the direction of [theta] increasing), we have

[delta]r dr r[delta][theta] d[theta] u = lim.-------- = --, v = lim. --------------- = r --------. (13) [delta]t dt [delta]t dt

Again, the velocities parallel and perpendicular to OP change in the time [delta]t from u, v to u - v[delta][theta], v + u[delta][theta], ultimately. The component accelerations at P in these directions are therefore

du d[theta] d²r /d[theta]\² \ -- - v -------- = --- - r ( -------- ), | dt dt dt² \ dt / | > (14) dv d[theta] 1 d / d[theta]\ | -- + u -------- = --- --- ( r² -------- ), | dt dt r dt \ dt / /

respectively.

In the case of a central force, with O as pole, the transverse acceleration vanishes, so that

r²d[theta]/dt = h, (15)

where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times. The radial resolution gives

d²r /d[theta]\² --- - r ( -------- ) = -P, (16) dt² \ dt /

where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by means of (15), we obtain the general differential equation of central orbits, viz.

d²u P --------- + u = ----. (17) d[theta]² h²u²

If, for example, the law be that of the inverse square, we have P = [mu]u², and the solution is of the form

[mu] u = ------ {1 + e cos ([theta] - [alpha])}, (18) h²

where e, [alpha] are arbitrary constants. This is recognized as the polar equation of a conic referred to the focus, the half latus-rectum being h²/[mu].

The law of the inverse cube P = [mu]u³ is interesting by way of contrast. The orbits may be divided into two classes according as h² <> [mu], i.e. according as the transverse velocity (hu) is greater or less than the velocity [root]([mu]·u) appropriate to a circular orbit at the same distance. In the former case the equation (17) takes the form

d²u -------- + m²u = 0, (19) d[theta]²

the solution of which is

au = sin m ([theta] - [alpha]). (20)

The orbit has therefore two asymptotes, inclined at an angle [pi]/m. In the latter case the differential equation is of the form

d²u --------- = m²u, (21) d[theta]²

so that

u = A e^(m[theta]) + B e^(-m[theta]) (22)

If A, B have the same sign, this is equivalent to

au = cosh m[theta], (23)

if the origin of [theta] be suitably adjusted; hence r has a maximum value [alpha], and the particle ultimately approaches the pole asymptotically by an infinite number of convolutions. If A, B have opposite signs the form is

au = sinh m[theta], (24)

this has an asymptote parallel to [theta] = 0, but the path near the origin has the same general form as in the case of (23). If A or B vanish we have an equiangular spiral, and the velocity at infinity is zero. In the critical case of h² = [mu], we have d²u/d[theta]² = 0, and

u = A[theta] + B; (25)

the orbit is therefore a "reciprocal spiral," except in the special case of A = 0, when it is a circle. It will be seen that unless the conditions be exactly adjusted for a circular orbit the particle will either recede to infinity or approach the pole asymptotically. This problem was investigated by R. Cotes (1682-1716), and the various curves obtained arc known as _Coles's spirals_.

A point on a central orbit where the radial velocity (dr/dt) vanishes is called an _apse_, and the corresponding radius is called an _apse-line_. If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed. It follows that the angle between successive apse-lines is constant; it is called the _apsidal angle_ of the orbit.

If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5), _ h² / [oo] -- = 2 | P dr; (26) p² _/ r

this determines the form of the critical orbit, as it is called. If P = [mu]/r^[n], its polar equation is

r^m cos m[theta] = a^m, (27)

where m = ½(3 - n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2 gives the parabola as before.

If we eliminate d[theta]/dt between (15) and (16) we obtain

d²r h² --- - -- = -P = -f(r), dt² r³

say. We may apply this to the investigation of the stability of a circular orbit. Assuming that r = a + x, where x is small, we have, approximately,

d²x h² / 3x\ --- - -- ( 1 - -- ) = -f(a) - xf´(a). dt² r³ \ a /

Hence if h and a be connected by the relation h² = a³f(a) proper to a circular orbit, we have _ _ d²x | 3 | --- + | f´(a) + --- f(a)| x = 0. (28) dt² |_ a _|

If the coefficient of x be positive the variations of x are simple-harmonic, and x can remain permanently small; the circular orbit is then said to be stable. The condition for this may be written _ _ d | | -- | a³f(a) | > 0, (29) da |_ _|

i.e. the intensity of the force in the region for which r = a, nearly, must diminish with increasing distance less rapidly than according to the law of the inverse cube. Again, the half-period of x is [pi]/sqrt[f´(a) + 3^{-1}f(a)], and since the angular velocity in the orbit is h/a², approximately, the apsidal angle is, ultimately, _ _ / | f(a) | [pi] / | --------------- |, (30) \/ |_ af´(a) + 3f(a) _|

or, in the case of f(a) = [mu]/r^n, [pi]/[root](3 - n). This is in agreement with the known results for n = 2, n = -1.

We have seen that under the law of the inverse square all finite orbits are elliptical. The question presents itself whether there then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves. If this is the case, the apsidal angle must evidently be commensurable with [pi], and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant. Equating the expression (30) to [pi]/m, we find that f(a) = C/a^n, where n = 3 - m². The force must therefore vary as a power of the distance, and n must be less than 3. Moreover, the case n = 2 is the only one in which the critical orbit (27) can be regarded as the limiting form of a closed curve. Hence the only law of force which satisfies the conditions is that of the inverse square.

At the beginning of § 13 the velocity of a moving point P was represented by a vector [->OV] drawn from a fixed origin O. The locus of the point V is called the _hodograph_ (q.v.); and it appears that the velocity of the point V along the hodograph represents in magnitude and in direction the acceleration in the original orbit. Thus in the case of a plane orbit, if v be the velocity of P, [psi] the inclination of the direction of motion to some fixed direction, the polar co-ordinates of V may be taken to be v, [psi]; hence the velocities of V along and perpendicular to OV will be dv/dt and vd[psi]/dt. These expressions therefore give the tangential and normal accelerations of P; cf. § 13 (12).

In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity. In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit. In the case of a central orbit described under the law of the inverse square we have v = h/SY = h. SZ/b², where S is the centre of force, SY is the perpendicular to the tangent at P, and Z is the point where YS meets the auxiliary circle again. Hence the hodograph is similar and similarly situated to the locus of Z (the auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case. The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation. The pole O of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola. In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the "reciprocal polar" of the orbit with respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole, described with constant velocity.

§ 15. _Kinetics of a System of Discrete Particles._--The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (§ 8). Thus taking any point O as base, we have first a _linear momentum_ whose components referred to rectangular axes through O are

[Sigma](m[.x]), [Sigma](m[.y]), [Sigma](m[.z]); (1)

its representative vector is the same whatever point O be chosen. Secondly, we have an _angular momentum_ whose components are

[Sigma]{m(y[.z] - z[.y])}, [Sigma]{m(z[.x] - xz[.z])}, [Sigma]{m(x[.y] - y[.x])}, (2)

these being the sums of the moments of the momenta of the several particles about the respective axes. This is subject to the same relations as a couple in statics; it may be represented by a vector which will, however, in general vary with the position of O.

The linear momentum is the same as if the whole mass were concentrated at the centre of mass G, and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t + [delta]t. Thus

__ / [->PP] \ __ [->GG´] \ ( m·-------- ) = \ (m)·--------. (3) /__ \ [delta]t / /__ [delta]t

Analytically we have

d d[|x] [Sigma](m[.x]) = --- [Sigma](mx) = [Sigma](m)·-----. (4) dt dt

with two similar formulae.

Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute motion is the same as the angular momentum of the motion relative to G. For the velocity of a particle m at P may be replaced by two components one of which (v) is identical in magnitude and direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the components mv of momentum is equivalent to a single localized vector [Sigma](m)·v in a line through G, and has therefore zero moment about any axis through G; hence in taking moments about such an axis we need only regard the velocities relative to G. In symbols, we have

/ d[|z] d[|y]\ [Sigma]{m(y[.z] - z[.y])} = [Sigma](m)·( y ----- - z ----- ) + [Sigma]{m([eta][zeta] - [.zeta][eta])}. (5) \ dt dt /

since [Sigma](m[xi]) = 0, [Sigma](m[xi]) = 0, and so on, the notation being as in § 11. This expresses that the moment of momentum about any fixed axis (e.g. Ox) is equal to the moment of momentum of the motion relative to G about a parallel axis through G, together with the moment of momentum of the whole mass supposed concentrated at G and moving with this point. If in (5) we make O coincide with the instantaneous position of G, we have [|x], [|y], [|z] = 0, and the theorem follows.

Finally, the rates of change of the components of the angular momentum of the motion relative to G referred to G as a moving base, are equal to the rates of change of the corresponding components of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let G´ be a consecutive position of G. At the instant t + [delta]t the momenta of the system are equivalent to a linear momentum represented by a localized vector [Sigma](m)·(v + [delta]v) in a line through G´ tangential to the path of G´, together with a certain angular momentum. Now the moment of this localized vector with respect to any axis through G is zero, to the first order of [delta]t, since the perpendicular distance of G from the tangent line at G´ is of the order ([delta]t)². Analytically we have from (5),

d / d[|z]² d²[|y] \ d --- [Sigma] {m (y[.z] - z[.y])} = [Sigma](m)·( y ------ - z ------- ) + --- [Sigma] {m([eta][zeta - [zeta][.eta])} (6) dt \ dt² dt² / dt

If we put x, y, z = 0, the theorem is proved as regards axes parallel to Ox.

Next consider the kinetic energy of the system. If from a fixed point O we draw vectors [->OV1], [->OV2] to represent the velocities of the several particles m1, m2 ..., and if we construct the vector

[Sigma](m·[->OV]) [->OK] = ----------------- (7) [Sigma](m)

this will represent the velocity of the mass-centre, by (3). We find, exactly as in the proof of Lagrange's First Theorem (§ 11), that

½[Sigma](m·OV²) = ½[Sigma](m)·OK² + ½[Sigma](m·KV²); (8)

i.e. the total kinetic energy is equal to the kinetic energy of the whole mass supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G. The latter may be called the _internal kinetic energy_ of the system. Analytically we have _ _ | /d[|x]\² /d[|y]\² /d[|z]\ | ½[Sigma]{m([.x]² + [.y]² + [.z]²)} = ½[Sigma](m)·| ( ----- ) + ( ----- ) + ( ----- ) | |_ \ dt / \ dt / \ dt / _|

+ ½[Sigma] {m([zeta]² + [.eta]² + [zeta]²)}. (9)

There is also an analogue to Lagrange's Second Theorem, viz.

[Sigma][Sigma] (m_p m_q·V_p V_q²) ½[Sigma](m·KV²) = ½ --------------------------------- (10) [Sigma]m

which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of particles. This formula is due to Möbius.

The preceding theorems are purely kinematical. We have now to consider the effect of the forces acting on the particles. These may be divided into two categories; we have first, the _extraneous forces_ exerted on the various particles from without, and, secondly, the mutual or _internal forces_ between the various pairs of particles. It is assumed that these latter are subject to the law of equality of action and reaction. If the equations of motion of each particle be formed separately, each such internal force will appear twice over, with opposite signs for its components, viz. as affecting the motion of each of the two particles between which it acts. The full working out is in general difficult, the comparatively simple problem of "three bodies," for instance, in gravitational astronomy being still unsolved, but some general theorems can be formulated.

The first of these may be called the _Principle of Linear Momentum_. If there are no extraneous forces, the resultant linear momentum is constant in every respect. For consider any two particles at P and Q, acting on one another with equal and opposite forces in the line PQ. In the time [delta]t a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant linear momentum of the system is unaltered. If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric addition of the total impulse of the extraneous forces. It follows, by the preceding kinematic theory, that the mass-centre G of the system will move exactly as if the whole mass were concentrated there and were acted on by the extraneous forces applied parallel to their original directions. For example, the mass-centre of a system free from extraneous force will describe a straight line with constant velocity. Again, the mass-centre of a chain of particles connected by strings, projected anyhow under gravity, will describe a parabola.

The second general result is the _Principle of Angular Momentum_. If there are no extraneous forces, the moment of momentum about any fixed axis is constant. For in time [delta]t the mutual action between two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have equal and opposite moments about the fixed axis. If extraneous forces act, the total angular momentum about any fixed axis is in time [delta]t increased by the total extraneous impulse about that axis. The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time [delta]t we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G. If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect. A plane through G perpendicular to this vector has a fixed direction in space, and is called the _invariable plane_; it may sometimes be conveniently used as a plane of reference.

For example, if we have two particles connected by a string, the invariable plane passes through the string, and if [omega] be the angular velocity in this plane, the angular momentum relative to G is

m1[omega]1r1·r1 + m2[omega]r2·r2 = (m1r1² + m2r2²)[omega],

where r1, r2 are the distances of m1, m2 from their mass-centre G. Hence if the extraneous forces (e.g. gravity) have zero moment about G, [omega] will be constant. Again, the tension R of the string is given by

m1m2 R = m1[omega]²r1 = ------- [omega]²a, m1 + m2

where a = r1 + r2. Also by (10) the internal kinetic energy is

m1m2 ½ ------- [omega]²a². m1 + m2

The increase of the kinetic energy of the system in any interval of time will of course be equal to the total work done by all the forces acting on the particles. In many questions relating to systems of discrete particles the internal force R_pq (which we will reckon positive when attractive) between any two particles m_p, m_q is a function only of the distance r_pq between them. In this case the work done by the internal forces will be represented by _ / -[Sigma] | R_(pg) dr_(pq), _/

when the summation includes every pair of particles, and each integral is to be taken between the proper limits. If we write _ / V = [Sigma] | R_(pq) dr_(pq), (11) _/

when r_pq ranges from its value in some standard configuration A of the system to its value in any other configuration P, it is plain that V represents the work which would have to be done in order to bring the system from rest in the configuration A to rest in the configuration P. Hence V is a definite function of the configuration P; it is called the _internal potential energy_. If T denote the kinetic energy, we may say then that the sum T + V is in any interval of time increased by an amount equal to the work done by the extraneous forces. In particular, if there are no extraneous forces T + V is constant. Again, if some of the extraneous forces are due to a conservative field of force, the work which they do may be reckoned as a diminution of the potential energy relative to the field as in § 13.

§ 16. _Kinetics of a Rigid Body. Fundamental Principles._--When we pass from the consideration of discrete particles to that of continuous distributions of matter, we require some physical postulate over and above what is contained in the Laws of Motion, in their original formulation. This additional postulate may be introduced under various forms. One plan is to assume that any body whatever may be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction. In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as in § 15.

An alternative procedure is to adopt the principle first formally enunciated by J. Le R. d'Alembert and since known by his name. If x, y, z be the rectangular co-ordinates of a mass-element m, the expressions m[:x], m[:y], m[:z] must be equal to the components of the total force on m, these forces being partly extraneous and partly forces exerted on m by other mass-elements of the system. Hence (m[:x], m[:y], m[:z]) is called the actual or _effective_ force on m. According to d'Alembert's formulation, the extraneous forces together with the _effective forces reversed_ fulfil the statical conditions of equilibrium. In other words, the whole assemblage of effective forces is statically equivalent to the extraneous forces. This leads, by the principles of § 8, to the equations

[Sigma](m[:x]) = X, [Sigma](m[:y]) = Y, [Sigma](m[:z]) = Z, \ > (1) [Sigma]{m(y[:z] - z[:y]) = L, [Sigma]{m(z[:x] - x[:z]) = M, [Sigma]{m(x[:y] - y[:x]) = N, /

where (X, Y, Z) and (L, M, N) are the force--and couple--constituents of the system of extraneous forces, referred to O as base, and the summations extend over all the mass-elements of the system. These equations may be written

d d d --- [Sigma](m[.x]) = X, --- [Sigma](m[.y]) = Y, --- [Sigma](m[.z]) = Z, \ dt dt dt | } (2) > (2) d d d | --- [Sigma]{m(y[.z] - z[.y]) = L, --- [Sigma]{m(z[.x]-x[.z]) = M, --- [Sigma]{m(x[.y] - y[.x]) = N, / dt dt dt

and so express that the rate of change of the linear momentum in any fixed direction (e.g. that of Ox) is equal to the total extraneous force in that direction, and that the rate of change of the angular momentum about any fixed axis is equal to the moment of the extraneous forces about that axis. If we integrate with respect to t between fixed limits, we obtain the principles of linear and angular momentum in the form previously given. Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies; they are assumed to hold for all material systems whatever. The peculiar status of rigid bodies is that the principles in question are in most cases sufficient for the complete determination of the motion, the dynamical equations (1 or 2) being equal in number to the degrees of freedom (six) of a rigid solid, whereas in cases where the freedom is greater we have to invoke the aid of other supplementary physical hypotheses (cf. ELASTICITY; HYDROMECHANICS).

The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done by the extraneous forces acting on the body. This is an immediate consequence of the fundamental postulate, in either of the forms above stated, since the internal forces do on the whole no work. The statement may be extended to a system of rigid bodies, provided the mutual reactions consist of the stresses in inextensible links, or the pressures between smooth surfaces, or the reactions at rolling contacts (§ 9).

§ 17. _Two-dimensional Problems._--In the case of rotation about a fixed axis, the principles take a very simple form. The position of the body is specified by a single co-ordinate, viz. the angle [theta] through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then d[theta]/dt, = [omega] say, is the _angular velocity_ of the body. The angular momentum of a particle m at a distance r from the axis is m[omega]r·r, and the total angular momentum is [Sigma](mr²)·[omega], or I[omega], if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the extraneous forces about the axis, we have

d --- (I[omega]) = N. (1) dt

This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation. If N = 0, [omega] is constant.

As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig. 72). Neglecting frictional resistance we have, if R be the tension of the string,

I[.omega] = Rb, m[.u] = mg - R,

whence mb² b[.omega] = ------- (2) 1 + mb²

This gives the acceleration of m as modified by the inertia of the wheel.

A "compound pendulum" is a body of any form which is free to rotate about a fixed horizontal axis, the only extraneous force (other than the pressures of the axis) being that of gravity. If M be the total mass, k the radius of gyration (§ 11) about the axis, we have

d / d[theta]\ --- ( Mk² -------- ) = -Mgh sin [theta], (3) dt \ dt /

where [theta] is the angle which the plane containing the axis and the centre of gravity G makes with the vertical, and h is the distance of G from the axis. This coincides with the equation of motion of a simple pendulum [§ 13 (15)] of length l, provided l = k²/h. The plane of the diagram (fig. 73) is supposed to be a plane through G perpendicular to the axis, which it meets in O. If we produce OG to P, making OP = l, the point P is called the _centre of oscillation_; the bob of a simple pendulum of length OP suspended from O will keep step with the motion of P, if properly started. If [kappa] be the radius of gyration about a parallel axis through G, we have k² = [kappa]² + h² by § 11 (16), and therefore l = h + [kappa]²/h, whence

GO·GP = [kappa]². (4)

This shows that if the body were swung from a parallel axis through P the new centre of oscillation would be at O. For different parallel axes, the period of a small oscillation varies as [root]l, or [root](GO + OP); this is least, subject to the condition (4), when GO = GP = [kappa]. The reciprocal relation between the centres of suspension and oscillation is the basis of Kater's method of determining g experimentally. A pendulum is constructed with two parallel knife-edges as nearly as possible in the same plane with G, the position of one of them being adjustable. If it could be arranged that the period of a small oscillation should be exactly the same about either edge, the two knife-edges would in general occupy the positions of conjugate centres of suspension and oscillation; and the distances between them would be the length l of the equivalent simple pendulum. For if h1 + [kappa]²/h1 = h2 + [kappa]²/h2, then unless h1 = h2, we must have [kappa]² = h1h2, l = h1 + h2. Exact equality of the two observed periods ([tau]1, [tau]2, say) cannot of course be secured in practice, and a modification is necessary. If we write l1 = h1 + [kappa]²/h1, l2 = h2 + [kappa]²/h2, we find, on elimination of [kappa],

l1 + l2 l1 - l2 ½ ------- + ½ ------- = 1, h1 + h2 h1 - h2

whence

4[pi]² ½ ([tau]1² + [tau]2²) ½ ([tau]1² - [tau]2²) ------ = --------------------- + --------------------- (5) g h1 + h2 h1 - h2

The distance h1 + h2, which occurs in the first term on the right hand can be measured directly. For the second term we require the values of h1, h2 separately, but if [tau]1, [tau]2 are nearly equal whilst h1, h2 are distinctly unequal this term will be relatively small, so that an approximate knowledge of h1, h2 is sufficient.

As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position. The equation of motion is of the type

I[:theta] = -K[theta], (6)

and the period is therefore [tau] = 2[pi][root](I/K). If by the attachment of another body of known moment of inertia I´, the period is altered from [tau] to [tau]´, we have [tau]´ = 2[pi][root][(I + I´)/K]. We are thus enabled to determine both I and K, viz.

I/I´ = [tau]²/([tau]´² - [tau]²), K = 4[pi]²[tau]²I/([tau]´² - [tau]²). (7)

The couple may be due to the earth's magnetism, or to the torsion of a suspending wire, or to a "bifilar" suspension. In the latter case, the body hangs by two vertical threads of equal length l in a plane through G. The motion being assumed to be small, the tensions of the two strings may be taken to have their statical values Mgb/(a + b), Mga/(a + b), where a, b are the distances of G from the two threads. When the body is twisted through an angle [theta] the threads make angles a[theta]/l, b[theta]/l with the vertical, and the moment of the tensions about the vertical through G is accordingly -K[theta], where K = M gab/l.

For the determination of the motion it has only been necessary to use one of the dynamical equations. The remaining equations serve to determine the reactions of the rotating body on its bearings. Suppose, for example, that there are no extraneous forces. Take rectangular axes, of which Oz coincides with the axis of rotation. The angular velocity being constant, the effective force on a particle m at a distance r from Oz is m[omega]²r towards this axis, and its components are accordingly -[omega]²mx, -[omega]²my, O. Since the reactions on the bearings must be statically equivalent to the whole system of effective forces, they will reduce to a force (X Y Z) at O and a couple (L M N) given by

X = -[omega]²[Sigma](mx) = -[omega]²[Sigma](m)[|x], Y = -[omega]²[Sigma](my) = -[omega]²[Sigma](m)[|y], Z = 0,

L = [omega]²[Sigma](myz), M = -[omega]²[Sigma](mzx), N = 0, (8)

where [|x], [|y] refer to the mass-centre G. The reactions do not therefore reduce to a single force at O unless [Sigma](myz) = 0, [Sigma](msx) = 0, i.e. unless the axis of rotation be a principal axis of inertia (§ 11) at O. In order that the force may vanish we must also have x, y = 0, i.e. the mass-centre must lie in the axis of rotation. These considerations are important in the "balancing" of machinery. We note further that if a body be free to turn about a fixed point O, there are three mutually perpendicular lines through this point about which it can rotate steadily, without further constraint. The theory of principal or "permanent" axes was first investigated from this point of view by J. A. Segner (1755). The origin of the name "deviation moment" sometimes applied to a product of inertia is also now apparent.

Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle [theta] through which the body has turned from some standard position. The components of linear momentum are then M[.x], M[.y], and the angular momentum relative to G as base is I[.theta], where M is the mass and I the moment of inertia about G. If the extraneous forces be reduced to a force (X, Y) at G and a couple N, we have

M[:x] = X, M[:y] = Y, I[:theta] = N. (9)

If the extraneous forces have zero moment about G the angular velocity [.theta] is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant angular velocity, whilst its centre describes a parabola.

We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal on a plane of inclination [alpha]. If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have

M[:x] = Mg sin [alpha] - F, 0 = Mg cos [alpha] - R, M[kappa]²[:theta] = Fa (10)

where [kappa] is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig. 74. We have also the kinematical relation [.x] = a[.theta]. Hence

a² [kappa]² [:x] = ------------- g sin [alpha], R = Mg cos [alpha], F = ------------- Mg sin [alpha]. (11) [kappa]² + a² [kappa]² + a²

The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a²/([kappa]² + a²). For a homogeneous sphere this ratio is 5/7, for a uniform circular cylinder or disk 2/3, for a circular hoop or a thin cylindrical shell ½.

The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions. It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have

M([.x][:x] + [.y][:]y) + l[.theta][:theta] + X[.x] + Y[.y] + N[.theta], (12)

whence, integrating with respect to t,

½ M([.x]² + [.y]²) + ½I[.theta]² = [int](X dx + Y dy + Nd[theta]) + const. (13)

The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.

The formula (13) may be easily verified in the case of the compound pendulum, or of the solid rolling down an incline. As another example, suppose we have a circular cylinder whose mass-centre is at an excentric point, rolling on a horizontal plane. This includes the case of a compound pendulum in which the knife-edge is replaced by a cylindrical pin. If [alpha] be the radius of the cylinder, h the distance of G from its axis (O), [kappa] the radius of gyration about a longitudinal axis through G, and [theta] the inclination of OG to the vertical, the kinetic energy is 1/2M[kappa]²[.theta]² + ½M·CG²·[.theta]², by § 3, since the body is turning about the line of contact (C) as instantaneous axis, and the potential energy is--Mgh cos [theta]. The equation of energy is therefore

½ M([kappa]² + [alpha]² + h² - 2 ah cos [theta]) [.theta]² - Mgh cos [theta] - const. (14)

Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system of bodies) the configuration, the velocity of each particle at any instant will be proportional to [.q], and the total kinetic energy may be expressed in the form ½A[.q]², where A is in general a function of q [cf. equation (14)]. This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-ordinate q.

Thus in the case of a railway truck travelling with velocity u the kinetic energy is ½(M + m[kappa]²/[alpha]²)u², where M is the total mass, [alpha] the radius and [kappa] the radius of gyration of each wheel, and m is the sum of the masses of the wheels; the reduced inertia is therefore M + m[kappa]²/[alpha]². Again, take the system composed of the flywheel, connecting rod, and piston of a steam-engine. We have here a limiting case of three-bar motion (§ 3), and the instantaneous centre J of the connecting-rod PQ will have the position shown in the figure. The velocities of P and Q will be in the ratio of JP to JQ, or OR to OQ; the velocity of the piston is therefore y[.theta], where y = OR. Hence if, for simplicity, we neglect the inertia of the connecting-rod, the kinetic energy will be ½(I + My²)[.theta]², where I is the moment of inertia of the flywheel, and M is the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent moment of inertia of the flywheel by the variable amount My². If, on the other hand, we take OP (= x) as our variable, the kinetic energy is 1/2(M + I/y²)[.x]². We may also say, therefore, that the effect of the flywheel is to increase the apparent mass of the piston by the amount I/y²; this becomes infinite at the "dead-points" where the crank is in line with the connecting-rod.

If the system be "conservative," we have

½ Aq² + V = const., (15)

where V is the potential energy. If we differentiate this with respect to t, and divide out by [.q], we obtain

dA dV A[:q] + ½ -- q² + -- = 0 (16) dq dq

as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this must be satisfied by [.q] = O; this requires that dV/dq = 0, i.e. the potential energy must be "stationary." To examine the effect of a small disturbance from equilibrium we put V = f(q), and write q = q0 + [eta], where q0 is a root of f´(q0) = 0 and [eta] is small. Neglecting terms of the second order in [eta] we have dV/dq = f´(q) = f´´(q0)·[eta], and the equation (16) reduces to

A[:eta] + f´´(q0)[eta] = 0, (17)

where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if f´´(q0) > 0, i.e. if V is a minimum in the configuration of equilibrium, the variation of [eta] is simple-harmonic, and the period is 2[pi][root][A/f´´(q0)]. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If f´´(q0) < 0, the solution of (17) will involve real exponentials, and [eta] will in general increase until the neglect of the terms of the second order is no longer justified. The configuration q = q0, is then unstable.

As an example of the method, we may take the problem to which equation (14) relates. If we differentiate, and divide by [theta], and retain only the terms of the first order in [theta], we obtain

{x² + (h - [alpha])²} [:theta] + gh[theta] = 0, (18)

as the equation of small oscillations about the position [theta] = 0. The length of the equivalent simple pendulum is {[kappa]² + (h - [alpha])²}/h.

The equations which express the change of motion (in two dimensions) due to an instantaneous impulse are of the forms

M(u´- u) = [xi], M([nu]´ - [nu]) = [eta], I([omega]´ - [omega]) = [nu]. (19)

Here u´, [nu]´ are the values of the component velocities of G just before, and u, [nu] their values just after, the impulse, whilst [omega]´, [omega] denote the corresponding angular velocities. Further, [xi], [eta] are the time-integrals of the forces parallel to the co-ordinate axes, and [nu] is the time-integral of their moment about G. Suppose, for example, that a rigid lamina at rest, but free to move, is struck by an instantaneous impulse F in a given line. Evidently G will begin to move parallel to the line of F; let its initial velocity be u´, and let [omega]´ be the initial angular velocity. Then Mu´ = F, I[omega]´ = F·GP, where GP is the perpendicular from G to the line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be

u´ - [omega]´·GC = (F/M)·(I - GC·CP/[kappa]²),

where [kappa]² is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided GC·GP = [kappa]². If this condition be satisfied there would be no impulsive reaction at C even if this point were fixed. The point P is therefore called the _centre of percussion_ for the axis at C. It will be noted that the relation between C and P is the same as that which connects the centres of suspension and oscillation in the compound pendulum.

§ 18. _Equations of Motion in Three Dimensions._--It was proved in § 7 that a body moving about a fixed point O can be brought from its position at time t to its position at time t + [delta]t by an infinitesimal rotation [epsilon] about some axis through O; and the limiting position of this axis, when [delta]t is infinitely small, was called the "instantaneous axis." The limiting value of the ratio [epsilon]/[delta]t is called the _angular velocity_ of the body; we denote it by [omega]. If [xi], [eta], [zeta] are the components of [epsilon] about rectangular co-ordinate axes through O, the limiting values of [xi]/[delta]t, [eta]/[delta]t, [zeta]/[delta]t are called the _component angular velocities_; we denote them by p, q, r. If l, m, n be the direction-cosines of the instantaneous axis we have

p = l[omega], q = m[omega], r = n[omega], (1) p² + q² + r² = [omega]². (2)

If we draw a vector OJ to represent the angular velocity, then J traces out a certain curve in the body, called the _polhode_, and a certain curve in space, called the _herpolhode_. The cones generated by the instantaneous axis in the body and in space are called the polhode and herpolhode cones, respectively; in the actual motion the former cone rolls on the latter (§ 7).

The special case where both cones are right circular and [omega] is constant is important in astronomy and also in mechanism (theory of bevel wheels). The "precession of the equinoxes" is due to the fact that the earth performs a motion of this kind about its centre, and the whole class of such motions has therefore been termed _precessional_. In fig. 78, which shows the various cases, OZ is the axis of the fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode, which are of course both circles. If [alpha]be the semi-angle of the rolling cone, [beta] the constant inclination of OC to OZ, and [.psi] the angular velocity with which the plane ZOC revolves about OZ, then, considering the velocity of a point in OC at unit distance from O, we have

[omega] sin [alpha] = ±[.psi] sin [beta], (3)

where the lower sign belongs to the third case. The earth's precessional motion is of this latter type, the angles being [alpha] = .0087´´, [beta] = 23° 28´.

If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T is given by

2T = [Sigma] {m([omega]·PN)²} = [omega]²·[Sigma](m·PN²) = I[omega]², (4)

where I is the moment of inertia about the instantaneous axis. With the same notation for moments and products of inertia as in § 11 (38), we have

I = Al² + Bm² + Cn² - 2Fmn - 2Gnl - 2Hlm,

and therefore by (1),

2T = Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpq. (5)

Again, if x, y, z be the co-ordinates of P, the component velocities of m are

qz - ry, rx - pz, py - qx, (6)

by § 7 (5); hence, if [lambda], [mu], [nu] be now used to denote the component angular momenta about the co-ordinate axes, we have [lambda] = [Sigma][m(py - qx)y - m(rx - pz)z], with two similar formulae, or

[dP]T \ [lambda] = Ap - Hq - Gr= -----, | [dP]p | | [dP]T | [mu] = -Hp + Bq - Fr = -----, > (7) [dP]q | | [dP]T | [nu] = -Gp - Fq + Cr = -----. | [dP]r /

If the co-ordinate axes be taken to coincide with the principal axes of inertia at O, at the instant under consideration, we have the simpler formulae

2T = Ap² + Bq² + Cr², (8)

[lambda] = Ap, [mu] = Bq, [nu] = Cr. (9)

It is to be carefully noticed that the axis of resultant angular momentum about O does not in general coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or [lambda], [mu], [nu]. The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis. Again, if [Gamma] be the resultant angular momentum, so that

[lambda]² + [mu]² + [nu]² = [Gamma]², (10)

the length of the perpendicular OH on the tangent plane at J is

Ap p Bq q Cr r 2T [rho] OH = ------- · -------[rho] + ------- · -------[rho] + ------- · -------[rho] = ------- · -------, (11) [Gamma] [omega] [Gamma] [omega] [Gamma] [omega] [Gamma] [omega]

where [rho] = OJ. This relation will be of use to us presently (§ 19).

The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r. The component velocities of any point whose co-ordinates relative to O are x, y, z are then

u + qz - ry, v + rx - pz, w + py - qx (12)

by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the kinetic energy is given by

2T = M0(u² + v² + w²) + Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpg, (13)

where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-centre; cf. § 15 (9).

The components [xi], [eta], [zeta] of linear momentum are

[dP]T [dP]T [dP]T [xi] = M0u = -----, [eta] = M0v = -----, [zeta] = M0w = -----, (14) [dP]u [dP]v [dP]w

whilst those of the relative angular momentum are given by (7). The preceding formulae are sufficient for the treatment of instantaneous impulses. Thus if an impulse ([xi], [eta], [zeta], [lambda], [mu], [nu]) change the motion from (u, v, w, p, q, r) to (u´, v´, w´, p´, q´, r´) we have

M0(u´- u) = [xi], M0(v´- v) = [eta], M0(w´- w) = [zeta], \ > (15) A(p´ - p) = [lambda], B(q´- q) = [mu], C(r´- r) = [nu], /

where, for simplicity, the co-ordinate axes are supposed to coincide with the principal axes at the mass-centre. Hence the change of kinetic energy is

T´- T = [xi] · ½(u + u´) + [eta] · ½(v + v´) + [zeta] · ½(w + w´), + [lambda] · ½(p + p´) + [mu] · ½(q + q´) + [nu] · ½(r + r´). (16)

The factors of [xi], [eta], [zeta], [lambda], [mu], [nu] on the right-hand side are proportional to the constituents of a possible infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the work done by the given system of impulsive forces in such a displacement. As in § 9 this must be equal to the total work done in such a displacement by the several forces, whatever they are, which make up the impulse. We are thus led to the following statement: the change of kinetic energy due to any system of impulsive forces is equal to the sum of the products of the several forces into the semi-sum of the initial and final velocities of their respective points of application, resolved in the directions of the forces. Thus in the problem of fig. 77 the kinetic energy generated is ½M([kappa]² + Cq²)[omega]´², if C be the instantaneous centre; this is seen to be equal to ½F·[omega]´·CP, where [omega]´·CP represents the initial velocity of P.

The equations of continuous motion of a solid are obtained by substituting the values of [xi], [eta], [zeta], [lambda], [mu], [nu] from (14) and (7) in the general equations

d[xi] d[eta] d[zeta] \ ----- = X, ------ = Y, ------- = Z, | dt dt dt | > (17) d[lambda] d[mu] d[nu] | --------- = L, ----- = M, ----- = N, | dt dt dt /

where (X, Y, Z, L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction. The resulting equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in consequence of the changing orientation of the body with respect to the co-ordinate axes.

An exception occurs, however, in the case of a solid which is kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform sphere. The equations then take the forms

M0[.u] = X, M0[.v] = Y, M0[.w] = Z, C[.p] = L, C[.q] = M, C[.r] = N, (18)

where C is the constant moment of inertia about any axis through the mass-centre. Take, for example, the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to the plane, so that the equation of the latter is z = -a. We will suppose that the extraneous forces consist of a known force (X, Y, Z) at the centre, and of the reactions (F1, F2, R) at the point of contact. Hence

M0[.u] = X + F1, M0[.v] = Y + F2, 0 = Z + R, \ C[.p] = F2a, C[.q] = -F1a, C[.r] = 0. / (19)

The last equation shows that the angular velocity about the normal to the plane is constant. Again, since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the geometrical relations

u + qa = 0, v + pa = 0, w = 0, (20)

by (12). Eliminating p, q, we get

(M0 + Ca^-2)[.u] = X, (M0 + Ca^-2)[.v] = Y. (21)

The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the sphere were increased by C/[alpha]². Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a parabola with an acceleration

g sin [alpha]/(1 + C/Ma²)

parallel to the lines of greatest slope.

Take next the case of a sphere rolling on a fixed spherical surface. Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, z be the co-ordinates of this centre relative to axes through O, the centre of the fixed sphere. If the only extraneous forces are the reactions (P, Q, R) at the point of contact, we have

M0[:x] = P, M0[.y] = Q, M0[:z] = R, \ | a a a > (22) Cp = ---(yR - zQ), C[.q] = ---(zP - xR), C[.r] = ---(xQ - yP), | c c c /

the standard case being that where the rolling sphere is outside the fixed surface. The opposite case is obtained by reversing the sign of a. We have also the geometrical relations

[.x] = (a/c)(qz - ry), [.y] = (a/c)(rx - pz), [.z] = (a/c)(py - gx), (23)

If we eliminate P, Q, R from (22), the resulting equations are integrable with respect to t; thus

M0a M0a p = - ---(y[.z] - z[.y]) + [alpha], q = - ---(z[.x] - x[.z]) + [beta], Cc Cc

M0a r = - ---(x[.y] - y[.x]) + [gamma], (24) Cc

where [alpha], [beta], [gamma] are arbitrary constants. Substituting in (23) we find

/ M0a²\ a / M0a²\ a ( 1 + ---- )[.x] = ---([beta]z - [gamma]y), ( 1 + ---- )[.y] = ---([gamma]x - [alpha]z), \ C / c \ C / c

/ M0a²\ a ( 1 + ---- )[.z] = ---([alpha]y - [beta]x). (25) \ C / c

Hence [alpha][.x] + [beta][.y] + [gamma][.z] = 0, or

[alpha]x + [beta]y + [gamma]z = const.; (26)

which shows that the centre of the rolling sphere describes a circle. If the axis of z be taken normal to the plane of this circle we have [alpha] = 0, [beta] = 0, and

/ M0a²\ a / M0a²\ a ( 1 + ---- )[.x] = -[gamma]--- y, ( 1 + ----- )[.y] = [gamma]--- x. (27) \ C / c \ C / c

The solution of these equations is of the type

x = b cos ([sigma][tau] + [epsilon]), y = b sin ([sigma][iota] + [epsilon]), (28)

where b, [epsilon] are arbitrary, and

[gamma]a/c [sigma]= ---------- (29) 1 + M0a²/C

The circle is described with the constant angular velocity [sigma].

When the gravity of the rolling sphere is to be taken into account the preceding method is not in general convenient, unless the whole motion of G is small. As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the angular velocity n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory. If Oz be drawn vertically upwards, then in the beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small. Hence, omitting terms of the second order, we find

M0[:x] = P, M0[.y] = Q, R = M0g, \ > (30) C[.p] = -(M0ga/c)y + aQ, C[.q] = (M0ga/c)x - aP, C[.r] = 0. /

The last equation shows that the component r of the angular velocity retains (to the first order) the constant value n. The geometrical relations reduce to

[.x] = aq - (na/c)y, [.y] = -ap + (na/c)x. (31)

Eliminating p, g, P, Q, we obtain the equations

(C + M0a²)[:x] + (Cna/c)y - (M0ga²/c)x = 0, } (C + M0a²)[:y] - (Cna/c)x - (M0ga²/c)y = 0, } (32)

which are both contained in _ _ | d² Cna d M0ga² | |(C + M0a²)--- - i --- --- - ----- | (x + iy) = 0. (33) |_ dt² c dt c _|

This has two solutions of the type x + iy = [alpha]e^{i([sigma]t + [epsilon])}, where [alpha], [epsilon] are arbitrary, and [sigma] is a root of the quadratic

(C + M0a²)[sigma]² - (Cna/c)[sigma] + M0ga²/c = 0. (34)

If

n² > (4Mgc/C) (1 + M0a²/C), (35)

both roots are real, and have the same sign as n. The motion of G then consists of two superposed circular vibrations of the type

x = [alpha] cos ([sigma]t + [epsilon]), y = [alpha] sin ([sigma]t + [epsilon]), (36)

in each of which the direction of revolution is the same as that of the initial spin of the sphere. It follows therefore that the original position is stable provided the spin n exceed the limit defined by (35). The case of a sphere spinning about a vertical axis at the lowest point of a spherical bowl is obtained by reversing the signs of [alpha] and c. It appears that this position is always stable.

It is to be remarked, however, that in the first form of the problem the stability above investigated is practically of a limited or temporary kind. The slightest frictional forces--such as the resistance of the air--even if they act in lines through the centre of the rolling sphere, and so do not directly affect its angular momentum, will cause the centre gradually to descend in an ever-widening spiral path.

§ 19. _Free Motion of a Solid._--Before proceeding to further problems of motion under extraneous forces it is convenient to investigate the free motion of a solid relative to its mass-centre O, in the most general case. This is the same as the motion about a fixed point under the action of extraneous forces which have zero moment about that point. The question was first discussed by Euler (1750); the geometrical representation to be given is due to Poinsot (1851).

The kinetic energy T of the motion relative to O will be constant. Now T = ½I[omega]², where [omega] is the angular velocity and I is the moment of inertia about the instantaneous axis. If [rho] be the radius-vector OJ of the momental ellipsoid

Ax² + By² + Cz² = M[epsilon]^4 (1)

drawn in the direction of the instantaneous axis, we have I = M[epsilon]^4/[rho]² (§ 11); hence [omega] varies as [rho]. The locus of J may therefore be taken as the "polhode" (§ 18). Again, the vector which represents the angular momentum with respect to O will be constant in every respect. We have seen (§ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that

2T [rho] OH = ------- · -------, (2) [Gamma] [omega]

where [Gamma] is the resultant angular momentum about O. Since [omega] varies as [rho], it follows that OH is constant, and the tangent plane at J is therefore fixed in space. The motion of the body relative to O is therefore completely represented if we imagine the momental ellipsoid at O to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact. The fixed plane is parallel to the invariable plane at O, and the line OH is called the _invariable line_. The trace of the point of contact J on the fixed plane is the "herpolhode."

If p, q, r be the component angular velocities about the principal axes at O, we have

(A²p² + B²q² + C²r²)/[Gamma]² = (Ap² + Bq² + Cr²)/2T, (3)

each side being in fact equal to unity. At a point on the polhode cone x : y : z = p : q : r, and the equation of this cone is therefore

/ [Gamma]²\ / [Gamma]²\ / [Gamma]²\ A²( 1 - -------- )x² + B²( 1 - -------- )y² + C²( 1 - -------- )z² = 0. (4) \ 2AT / \ 2BT / \ 2CT /

Since 2AT - [Gamma]² = B (A - B)q² + C(A - C)r², it appears that if A > B > C the coefficient of x² in (4) is positive, that of z² is negative, whilst that of y² is positive or negative according as 2BT <> [Gamma]². Hence the polhode cone surrounds the axis of greatest or least moment according as 2BT <> [Gamma]². In the critical case of 2BT = [Gamma]² it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a re-entrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B + C > A, the curve has no points of inflexion. The invariable line OH describes another cone in the body, called the _invariable cone_. At any point of this we have x : y : z = Ap. Bq : Cr, and the equation is therefore

/ [Gamma]²\ / [Gamma]²\ / [Gamma]²\ ( 1 - -------- )x² + ( 1 - -------- )y² + ( 1 - -------- )z² = 0. (5) \ 2AT / \ 2BT / \ 2CT /

The signs of the coefficients follow the same rule as in the case of (4). The possible forms of the invariable cone are indicated in fig. 80 by means of the intersections with a concentric spherical surface. In the critical case of 2BT = [Gamma]² the cone degenerates into two planes. It appears that if the body be sightly disturbed from a state of rotation about the principal axis of greatest or least moment, the invariable cone will closely surround this axis, which will therefore never deviate far from the invariable line. If, on the other hand, the body be slightly disturbed from a state of rotation about the mean axis a wide deviation will take place. Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable. The question is greatly simplified when two of the principal moments are equal, say A = B. The polhode and herpolhode cones are then right circular, and the motion is "precessional" according to the definition of § 18. If [alpha] be the inclination of the instantaneous axis to the axis of symmetry, [beta] the inclination of the latter axis to the invariable line, we have

[Gamma] cos [beta] = C [omega] cos [alpha], [Gamma] sin [beta] = A [omega] sin [alpha], (6)

whence

A tan [beta] = --- tan [alpha]. (7) C

Hence [beta] <> [alpha], and the circumstances are therefore those of the first or second case in fig. 78, according as A <> C. If [psi] be the rate at which the plane HOJ revolves about OH, we have

sin [alpha] C cos [alpha] [psi] = ----------- [omega] = ------------- [omega], (8) sin [beta] A cos [beta]

by § 18 (3). Also if [.chi] be the rate at which J describes the polhode, we have [.psi] sin ([beta]-[alpha]) = [.chi] sin [beta], whence

sin([alpha] - [beta]) [.chi] = --------------------- [omega]. (9) sin[alpha]

If the instantaneous axis only deviate slightly from the axis of symmetry the angles [alpha], [beta] are small, and [.chi] = (A - C)A·[omega]; the instantaneous axis therefore completes its revolution in the body in the period

2[pi] A - C ------ = ----- [omega]. (10) [.chi] A

In the case of the earth it is inferred from the independent phenomenon of luni-solar precession that (C - A)/A = .00313. Hence if the earth's axis of rotation deviates slightly from the axis of figure, it should describe a cone about the latter in 320 sidereal days. This would cause a periodic variation in the latitude of any place on the earth's surface, as determined by astronomical methods. There appears to be evidence of a slight periodic variation of latitude, but the period would seem to be about fourteen months. The discrepancy is attributed to a defect of rigidity in the earth. The phenomenon is known as the _Eulerian nutation_, since it is supposed to come under the free rotations first discussed by Euler.

§ 20. _Motion of a Solid of Revolution._--In the case of a solid of revolution, or (more generally) whenever there is kinetic symmetry about an axis through the mass-centre, or through a fixed point O, a number of interesting problems can be treated almost directly from first principles. It frequently happens that the extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant. For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.

Suppose, for example, that a flywheel is rotating with angular velocity n about its axis, which is (say) horizontal, and that this axis is made to rotate with the angular velocity [psi] in the horizontal plane. The components of angular momentum about the axis of the flywheel and about the vertical will be Cn and A [psi] respectively, where A is the moment of inertia about any axis through the mass-centre (or through the fixed point O) perpendicular to that of symmetry. If [->OK] be the vector representing the former component at time t, the vector which represents it at time t + [delta]t will be [->OK´], equal to [->OK] in magnitude and making with it an angle [delta][psi]. Hence [->KK´] ( = Cn [delta][psi]) will represent the change in this component due to the extraneous forces. Hence, so far as this component is concerned, the extraneous forces must supply a couple of moment Cn[.psi] in a vertical plane through the axis of the flywheel. If this couple be absent, the axis will be tilted out of the horizontal plane in such a sense that the direction of the spin n approximates to that of the azimuthal rotation [.psi]. The remaining constituent of the extraneous forces is a couple A[:psi] about the vertical; this vanishes if [.psi] is constant. If the axis of the flywheel make an angle [theta] with the vertical, it is seen in like manner that the required couple in the vertical plane through the axis is Cn sin [theta] [.psi]. This matter can be strikingly illustrated with an ordinary gyroscope, e.g. by making the larger movable ring in fig. 37 rotate about its vertical diameter.

If the direction of the axis of kinetic symmetry be specified by means of the angular co-ordinates [theta], [psi] of § 7, then considering the component velocities of the point C in fig. 83, which are [.theta] and sin [theta][.psi] along and perpendicular to the meridian ZC, we see that the component angular velocities about the lines OA´, OB´ are -sin [theta] [.psi] and [.theta] respectively. Hence if the principal moments of inertia at O be A, A, C, and if n be the constant angular velocity about the axis OC, the kinetic energy is given by

2T = A ([.theta]² + sin² [theta][.psi]²) + Cn². (1)

Again, the components of angular momentum about OC, OA´ are Cn, -A sin [theta] [.psi], and therefore the angular momentum ([mu], say) about OZ is

[mu] = A sin² [theta][.psi] + Cn cos [theta]. (2)

We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the direction OG. If [theta] is constant the points C, A´ will in time [delta]t come to positions C´´, A´´ such that CC´´ = sin [theta] [delta][psi], A´A´´ = cos [theta] [delta][psi], and the angular momentum about OB´ will become Cn sin [theta] [delta][psi] - A sin [theta] [.psi] · cos [theta] [delta][psi]. Equating this to Mgh sin [theta] [delta]t, and dividing out by sin [theta], we obtain

A cos [theta] [.psi]² - Cn[.psi] + Mgh = 0, (3)

as the condition in question. For given values of n and [theta] we have two possible values of [.psi] provided n exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with [root](4AMgh cos [theta]), one value of [.psi] is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos [theta] approximately. The absence of g from the latter expression indicates that the circumstances of the rapid precession are very nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.

Again, take the case of a circular disk rolling in steady motion on a horizontal plane. The centre O of the disk is supposed to describe a horizontal circle of radius c with the constant angular velocity [.psi], whilst its plane preserves a constant inclination [theta] to the horizontal. The components of the reaction of the horizontal lane will be Mc[.psi]² at right angles to the tangent line at the point of contact and Mg vertically upwards, and the moment of these about the horizontal diameter of the disk, which corresponds to OB´ in fig. 83, is Mc[.psi]². [alpha] sin [theta] - Mg[alpha] cos [theta], where [alpha] is the radius of the disk. Equating this to the rate of increase of the angular momentum about OB´, investigated as above, we find

/ a \ a² ( C + Ma² + A --- cos [theta] ) [.psi]² = Mg --- cot [theta], (4) \ c / c

where use has been made of the obvious relation n[alpha] = c[.psi]. If c and [theta] be given this formula determines the value of [psi] for which the motion will be steady.

In the case of the top, the equation of energy and the condition of constant angular momentum ([mu]) about the vertical OZ are sufficient to determine the motion of the axis. Thus, we have

½A ([.theta]² + sin² [theta][.psi]²) + ½Cn² + Mgh cos [theta] = const., (5)

A sin² [theta][.psi] + [nu] cos [theta] = [mu], (6)

where [nu] is written for Cn. From these [.psi] may be eliminated, and on differentiating the resulting equation with respect to t we obtain

([mu] - [nu] cos [theta])([mu] cos [theta] - [nu]) A[:theta] - -------------------------------------------------- - Mgh sin [theta] = 0. (7) A sin³ [theta]

If we put [:theta] = 0 we get the condition of steady precessional motion in a form equivalent to (3). To find the small oscillation about a state of steady precession in which the axis makes a constant angle [alpha] with the vertical, we write [theta] = [alpha] + [chi], and neglect terms of the second order in [chi]. The result is of the form

[:chi] + [sigma]²[chi] = 0, (8)

where

[sigma]² = {([mu] - [nu] cos [alpha])² + 2([mu] - [nu] cos [alpha])([mu] cos [alpha] - [nu]) cos [alpha] + ([mu] cos [alpha] - [nu])²} / A² sin^4 [alpha]. (9)

When [nu] is large we have, for the "slow" precession [sigma] = [nu]/A, and for the "rapid" precession [sigma] = A/[nu] cos [alpha] = [.psi], approximately. Further, on examining the small variation in [.psi], it appears that in a slightly disturbed slow precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be easily observed. In a slightly disturbed rapid precession the superposed vibration is elliptic-harmonic, with a period equal to that of the precession itself. The ratio of the axes of the ellipse is sec [alpha], the longer axis being in the plane of [theta]. The result is that the axis of the top describes a circular cone about a fixed line making a small angle with the vertical. This is, in fact, the "invariable line" of the free Eulerian rotation with which (as already remarked) we are here virtually concerned. For the more general discussion of the motion of a top see GYROSCOPE.

§ 21. _Moving Axes of Reference._--For the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.

We will assume for the present that the origin O is fixed. The moving axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The components of angular momentum about Ox, Oy, Oz will be denoted as usual by [lambda], [mu], [nu]. Now consider a system of fixed axes Ox´, Oy´, Oz´ chosen so as to coincide at the instant t with the moving system Ox, Oy, Oz. At the instant t + [delta]t, Ox, Oy, Oz will no longer coincide with Ox´, Oy´, Oz´; in particular they will make with Ox´ angles whose cosines are, to the first order, 1, -r[delta]t, q[delta]t, respectively. Hence the altered angular momentum about Ox´ will be [lambda] + [delta][lambda] + ([mu] + [delta][mu]) (-r[delta]t) + ([nu] + [delta][nu]) q[delta]t. If L, M, N be the moments of the extraneous forces about Ox, Oy, Oz this must be equal to [lambda] + L[delta]t. Hence, and by symmetry, we obtain

d[lambda] \ --------- - r[nu] + q[nu] = L, | dt | | d[mu] | ----- - p[nu] + r[lanbda] = M, > (1) dt | | d[nu] | ----- - q[lambda] + p[nu] = N. | dt /

These equations are applicable to any dynamical system whatever. If we now apply them to the case of a rigid body moving about a fixed point O, and make Ox, Oy, Oz coincide with the principal axes of inertia at O, we have [lambda], [mu], [nu] = Ap, Bq, Cr, whence

dp \ A -- - (B - C) qr = L, | dt | | dq | B -- - (C - A) rp = M, > (2) dt | | dr | C -- - (A - B) pq = N. | dt /

If we multiply these by p, q, r and add, we get

d --- · ½(Ap² + Bq² + Cr²) = Lp + Mq + Nr, (3) dt

which is (virtually) the equation of energy.

As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity [omega] about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining couple is

L = (C - B) mn[omega]², M = (A - C) nl[omega]², N = (B - A) lm[omega]². (4)

If we reverse the signs, we get the "centrifugal couple" exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).

If in (2) we put, L, M, N = O we get the case of free rotation; thus

dp \ A -- = (B - C) qr, | dt | | dq | B -- = (C - A) rp, > (5) dt | | dr | C -- = (A - B) pq. | dt /

These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap² + Bq² + Cr² and A²p² + B²q² + C²r² are both constant. The former is, in fact, equal to 2T, and the latter to [Gamma]², where T is the kinetic energy and [Gamma] the resultant angular momentum.

To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write _ / [phi] d[phi] u = | ------------, [Delta][phi] = [root](1 - k² sin² [phi]), _/ 0 [Delta][phi]

we have, in the notation of elliptic functions, [phi] = am u. If we assume

p = p0 cos am ([sigma]t + [epsilon]), q = q0sin am ([sigma]t + [epsilon]), r = r0[Delta] am ([sigma]t + [epsilon]), (7)

we find

[sigma]p0 [sigma]q0 k²[sigma]r0 [.p] = - --------- qr, [.q] = --------- rp, [.r] = - ----------- pq. (8) q0r0 r0p0 p0q0

Hence (5) will be satisfied, provided

-[sigma]p0 B - C [sigma]q0 C - A -k²[sigma]r0 A - B ---------- = -----, --------- = -----, ------------ = -----. (9) q0r0 A r0p0 B p0q0 C

These equations, together with the arbitrary initial values of p, q, r, determine the six constants which we have denoted by p0, q0, r0, k², [sigma], [epsilon]. We will suppose that A > B > C. From the form of the polhode curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must vanish periodically. If we adopt one of these epochs as the origin of t, we have [epsilon] = 0, and p0, r0 will become identical with the initial values of p, r. The conditions (9) then lead to

A(A - C) (A - C)(B - C) A(A - B) p0² q0² = -------- p0², [sigma]² = -------------- r0², k² = -------- · ---. (10) B(B - C) AB C(B - C) r0²

For a real solution we must have k² < 1, which is equivalent to 2BT > [Gamma]². If the initial conditions are such as to make 2BT < [Gamma]², we must interchange the forms of p and r in (7). In the present case the instantaneous axis returns to its initial position in the body whenever [phi] increases by 2[pi], i.e. whenever t increases by 4K/[sigma], when K is the "complete" elliptic integral of the first kind with respect to the modulus k.

The elliptic functions degenerate into simpler forms when k² = 0 or k² = 1. The former case arises when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k² = 1, we must have 2BT = [Gamma]². We have seen that the alternative 2BT <> [Gamma]² determines whether the polhode cone surrounds the principal axis of least or greatest moment. The case of 2BT = [Gamma]², exactly, is therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with the axis of mean moment or approaches it asymptotically.

When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are

d[xi] d[eta] d[zeta] ----- - r[eta] + q[zeta] = X, ------ - p[zeta] - r[chi] = Y, ------- - q[chi] + p[eta] = Z, (11) dt dt dt

d[lambda] d[mu] \ --------- - r[mu] + q[nu] - w[eta] + v[zeta] = L, ----- - p[nu] + r[lambda]- u[zeta] + w[xi] = M, | dt dt | > (12) d[nu] | ----- - q[lambda] + p[mu] - v[xi] + u[eta] = N. / dt

To prove these, we may take fixed axes O´x´, O´y´, O´z´ coincident with the moving axes at time t, and compare the linear and angular momenta [xi] + [delta][xi], [eta] + [delta][eta], [zeta] + [delta][zeta], [lambda] + [delta][lambda], [mu] + [delta][mu], [nu] + [delta][nu] relative to the new position of the axes, Ox, Oy, Oz at time t + [delta]t with the original momenta [xi], [eta], [zeta], [lambda], [mu], [nu] relative to O´x´, O´y´, O´z´ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the moving origin coincide always with the mass-centre, we have [xi], [eta], [zeta] = M0u, M0v, M0w, where M0 is the total mass, and the equations simplify.

When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of motion about a fixed point O; the angular co-ordinates [theta], [phi], [psi] of Euler may then be used for the purpose. Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their instantaneous positions are

p = [.theta] sin [phi] - sin [theta] cos [phi][.psi], \ q = [.theta] cos [phi] + sin [theta] sin [phi][.psi], > (13) r = [.phi] + cos [theta][.psi], /

by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by

2T = A([.theta] sin [phi] - sin [theta] cos [phi][.psi])² \ + B([.theta] cos [phi] + sin [theta] sin [theta][psi])² > (14) + C([.phi] + cos [theta][.psi])². /

If A = B this reduces to

2T = A([.theta]² + sin² [theta][.psi]²) + C([.phi] + cos [theta][.psi])²; (15)

cf. § 20 (1).

§ 22. _Equations of Motion in Generalized Co-ordinates._--Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but their number is determinate, and expresses the number of _degrees of freedom_ of the system. We denote these co-ordinates by q1, q2, ... q_n. It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the q's, varying in form (of course) from particle to particle. Hence the kinetic energy T is given by

__ 2T = \ {m([.x]² + [.y]² + [.z]²)} /__

= a11[.q]1² + a22[.q]2² + ... + 2a12[.q]1[.q]2 + ..., (1)

where _ _ __ | { / [dP]x \² / [dP]y \² / [dP]z \² } | \ a_rr = \ | m { ( ------- ) + ( ------- ) + ( ------- ) } |, | /__ |_ { \[dP]q_r/ \[dP]q_r/ \[dP]q_r/ } _| | _ _ > (2) __ | / [dP]x [dP]x [dP]y [dP]y [dP]z [dP]z \ | | a_rs = \ | m ( ------- ------- + ------- ------- + ------- ------- ) | = a_sr. | /__ |_ \[dP]q_r [dP]q_s [dP]q_r [dP]q_s [dP]q_r [dP]q_s/ _| /

Thus T is expressed as a homogeneous quadratic function of the quantities [.q]1, [.q]2, ... [.q]_n, which are called the _generalized components of velocity_. The coefficients a_rr, a_rs are called the coefficients of inertia; they are not in general constants, being functions of the q's and so variable with the configuration. Again, If (X, Y, Z) be the force on m, the work done in an infinitesimal change of configuration is

[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ... + Q_n[delta]q_n, (3)

where

/ [dP]x [dP]y [dP]z \ Q_r = [Sigma]( X------- + Y------- + Z------- ). (4) \ [dP]q_r [dP]q_r [dP]q_r /

The quantities Q_r are called the _generalized components of force_.

The equations of motion of m being

m[:x] = X, m[:y] = Y, m[:z] = Z, (5)

we have _ _ __ | / [dP]x [dP]y [dP]z \ | \ | m ( [:x]------- + [:y]------- + [:z]------- ) | = Q_r. (6) /__ |_ \ [dP]q_r [dP]q_r [dP]q_r / _|

Now

[dP]x [dP]x [dP]x [.x] = ------[.q]1 + ------[.q]2 + ... + -------[.q]_n, (7) [dP]q1 [dP]q2 [dP]q_n

whence

[dP][.x] [dP]x ---------- = -------. (8) [dP][.q]_r [dP]q_r

Also

d / [dP]x \ [dP]²x [dP]²x [dP]²x [dP]x -- ( ------- ) = ------------[.q]1 + -------------[.q]2 + ... + --------------[.q]_r = --------. (9) dt \[dP]q_r/ [dP]q1[dP]q_r [dP]q2[dP]q_r [dP]q_n[dP]q_r [dP]q_r

Hence

[dP]x d / [dP]x \ d / [dP]x \ d / [dP][.x] \ [dP][.x] [:x]------- = ---( [.x]------- ) - [.x]---( ------- ) = ---( [.x]---------- ) - [.x]--------. (10) [dP]q_r dt \ [dP]q_r/ dt \[dP]q_r/ dt \ [dP][.q]_r/ [dP]q_r

By these and the similar transformations relating to y and z the equation (6) takes the form

d / [dP]T \ [dP]T --- ( ---------- ) - ------ = Q_r. (11) dt \[dP][.q]_r/ [dP]q_r

If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated. The above proof was given by Sir W. R. Hamilton (1835). Lagrange's own proof will be found under DYNAMICS, § _Analytical_. In a conservative system free from extraneous force we have

[Sigma](X [delta]x + Y [delta]y + Z [delta]z) = -[delta]V, (12)

where V is the potential energy. Hence

[dP]V Q_r = - -------, (13) [dP]q_r

and

d / [dP]T \ [dP]T [dP]V --- ( ---------- ) - ----- = - -------. (14) dt \[dP][.q]_r/ Vq_r [dP]q_r

If we imagine any given state of motion ([.q]1, [.q]2 ... [.q]_n) through the configuration (q1, q2, ... q_n) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11) with respect to t over the infinitely short duration of the impulse

[dP]T ---------- = Q_r´, (15) [dP][.q]_r

where Q_r´ is the time integral of Q_r and so represents a _generalized component of impulse_. By an obvious analogy, the expressions [dP]T/[dP][.q]_r may be called the _generalized components of momentum_; they are usually denoted by p_r thus

p_r = [dP]T/[dP][.q]_r = a_(1r)[.q]1 + a_(2r)[.q]2 + ... + a_(nr)[.q]_n. (16)

Since T is a homogeneous quadratic function of the velocities [.q]1, [.q]2, ... [.q]_n, we have

[dP]T [dP]T [dP]T 2T = ---------[.q]1 + ---------[.q]2 + ... + ----------[.q]_n = p1[.q]2 + p2[.q]2 + ... + p_n[.q]_n. (17) [dP][.q]1 [dP][.q]2 [dP][.q]_n

Hence

dT 2-- = [.p]1[.q]1 + [.p]2[.q]2 + ... [.p]_n[.q]_n \ dt | | + [.p]1[:q]1 + [.p]2[:q]2 + ... + [.p]_n[:q]_n | | / [dP]T \ / [dP]T \ / [dP]T \ | = ( --------- + Q1 ) [.q]1 + ( --------- + Q2 ) [.q]2 + ... + ( ---------- + Q_n )[.q]_n > (18) \[dP][.q]1 / \[dP][.q]2 / \[dP][.q]_n / | | [dP]T [dP]T [dP]T | + ---------[:q]1 + ---------[:q]2 + ... ----------[:q]_n | [dP][.q]1 [dP][.q]2 [dP][.q]_n | | dT | = -- + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n, / dt

or

dT -- = Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (19) dt

This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces. In the case of a conservative system free from extraneous force it becomes the equation of energy

d --- (T + V) = 0, or T + V = const., (20) dt

in virtue of (13).

As a first application of Lagrange's formula (11) we may form the equations of motion of a particle in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, [theta] the angle which OP makes with a fixed direction OZ, [psi] the azimuth of the plane ZOP relative to some fixed plane through OZ. The displacements of P due to small variations of these co-ordinates are [dP]r along OP, r [delta][theta] perpendicular to OP in the plane ZOP, and r sin [theta] [delta][psi] perpendicular to this plane. The component velocities in these directions are therefore [.r], r[.theta], r sin [theta][.psi], and if m be the mass of a moving particle at P we have

2T = m([.r]² + r²[.theta]² + r² sin² [theta][.psi]²). (21)

Hence the formula (11) gives

m([:r] - r[.theta]² - r sin² [theta][.psi]²) = R, \ | d | ---(mr²[.theta]) - mr² · sin [theta] cos [theta][.psi]² = [Theta], > (22) dt | | d | ---(mr² sin² [theta][.psi]) = [Psi]. / dt

The quantities R, [Theta], [Psi] are the coefficients in the expression R [delta]r + [Theta] [delta][theta] + [Psi] [delta][psi] for the work done in an infinitely small displacement; viz. R is the radial component of force, [Theta] is the moment about a line through O perpendicular to the plane ZOP, and [Psi] is the moment about OZ. In the case of the spherical pendulum we have r = l, [Theta] = - mgl sin [theta], [Psi] = 0, if OZ be drawn vertically downwards, and therefore

g \ [:theta] - sin [theta] cos [theta][.psi]² = - --- sin [theta], | l > (23) | sin² [theta][.psi] = h, /

where h is a constant. The latter equation expresses that the angular momentum ml² sin² [theta][.psi] about the vertical OZ is constant. By elimination of [.psi] we obtain

g [:theta] - h² cos² [theta] / sin^3[theta] = - --- sin [theta]. (24) l

If the particle describes a horizontal circle of angular radius [alpha] with constant angular velocity [Omega], we have [.omega] = 0, h = [Omega]² sin [alpha], and therefore

g [Omega]² = --- cos [alpha], (25) l

as is otherwise evident from the elementary theory of uniform circular motion. To investigate the small oscillations about this state of steady motion we write [theta] = [alpha] + [chi] in (24) and neglect terms of the second order in [chi]. We find, after some reductions,

[:chi] + (1 + 3 cos² [alpha]) [Omega]²[chi] = 0; (26)

this shows that the variation of [chi] is simple-harmonic, with the period

2[pi]/[root](1 + 3 cos² [alpha])·[Omega]

As regards the most general motion of a spherical pendulum, it is obvious that a particle moving under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a vertical plane passing through O and a point where the motion is horizontal is evidently a plane of symmetry as regards the path. Hence the path will be confined between two horizontal circles which it touches alternately, and the direction of motion is never horizontal except at these circles. In the case of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a closer investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity 2/3 ab[Omega]/l², where a, b are the semi-axes.

To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle. If [lambda], [mu], [nu] be the components of momentum, we have

[dP]T \ [lambda]= ------------ = A[.theta], | [dP][.theta] | | [dP]T | [mu] = ---------- = A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta], > (27) [dP][.psi] | | [dP]T | [nu] = ---------- = C ([.theta] + cos [theta][.psi]). / [dP][.phi]

The meaning of these quantities is easily recognized; thus [lambda] is the angular momentum about a horizontal axis normal to the plane of [theta], [mu] is the angular momentum about the vertical OZ, and [nu] is the angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V = Mgh cos [theta], if OZ be drawn vertically upwards. Hence the equations (11) become

A[:theta] - A sin [theta] cos [theta][.psi]² + C([.phi] + cos [theta][.psi]) [.psi] sin [theta] = Mgh sin [theta], \ d/dt · {A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta]} = 0, > (28) d/dt · {C([.phi] + cos [theta][.psi])} = 0, /

of which the last two express the constancy of the momenta [mu], [nu]. Hence

A[:theta] - A sin [theta] cos [theta][.psi]² + [nu] sin [theta][.psi] = Mgh sin [theta], \ (29) A sin² [theta][.psi] + [nu] cos [theta] = [mu]. /

If we eliminate [.psi] we obtain the equation (7) of § 20. The theory of disturbed precessional motion there outlined does not give a convenient view of the oscillations of the axis about the vertical position. If [theta] be small the equations (29) may be written

[nu]²- 4AMgh \ [:theta] - [theta][.omega]² = - ------------[theta], > (30) 4A² | [theta]²[.omega] = const., /

where

[nu] [omega] = [psi] - ---- t. (31) 2A

Since [theta], [omega] are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves with constant angular velocity [nu]/2A, we see by comparison with § 14 (15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation [nu]/2A, provided [nu]² > 4AMgh. This gives (in essentials) the theory of the "gyroscopic pendulum."

§ 23. _Stability of Equilibrium. Theory of Vibrations._--If, in a conservative system, the configuration (q1, q2, ... q_n) be one of equilibrium, the equations (14) of § 22 must be satisfied by [.q]1, [.q]2 ... [.q]_n = 0, whence

[dP]V / [dP]q_r = 0. (1)

A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance. No such simple proof is available to show without qualification that the above condition is _necessary_. If, however, we recognize the existence of dissipative forces called into play by any motion whatever of the system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V must continually diminish so long as the velocities [.q]1, [.q]2, ... [.q]_n differ from zero. Hence if the system be started from rest in a configuration for which V is less than in the equilibrium configuration considered, this quantity must still further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).

In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized cc-ordinates q1, q2, ... q_n that they shall vanish in the configuration in question. The potential energy is then given with sufficient approximation by an expression of the form

2V = c11q1² + c22q2² + ... + 2c12q1q2 + ..., (2)

a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V is stationary. The coefficients c_rr, c_rs are called _coefficients of stability_. We may further treat the coefficients of inertia a_rr, a_rs of § 22 (1) as constants. The Lagrangian equations of motion are then of the type

a_(1r)[:q]1 + a_(2r)[:q]2 + ... + a_(nr)[:q]_n + c_(1r)q1 + c_(2r)q2 + ... + c_(nr)q_n = Q_r, (3)

where Q_r now stands for a component of extraneous force. In a _free oscillation_ we have Q1, Q2, ... Q_n = 0, and if we assume

q_r = A_r e^(i[sigma]^t), (4)

we obtain n equations of the type

(c_(1r) - [sigma]²a_(1r)) A1 + (c_(2r) - [sigma]²a_(2r)) A2 + ... + (c_(nr) - [sigma]²a_nr) A_n = 0. (5)

Eliminating the n - 1 ratios A1 : A2 : ... : A_n we obtain the determinantal equation

[Delta]([sigma]²) = 0, (6)

where

[Delta]([sigma]²) = | c11 - [sigma]²a11, c21 - [sigma]²a21, ..., C_(n1) - [sigma]²a_(nl) | | c12 - [sigma]²a12, c22 - [sigma]²a22, ..., C_(n2) - [sigma]²a_(n2) | | . . ... . | | . . ... . | (7) | . . ... . | | c_(1n) - [sigma]²a{1n}, c_(2n) - [sigma]²a_(2n), ..., C_(nn) - [sigma]²a_(nn) |

The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation in [sigma]² are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities A1, A2, ... A_n, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors of any one row in the determinate [Delta]([sigma]²). By combining the solutions corresponding to a pair of equal and opposite values of [sigma] we obtain a solution in real form:

q_r = C_(a_r) cos ([sigma]t + [epsilon]), (8)

where a1, a2 ... a_r are a determinate series of quantities having to one another the above-mentioned ratios, whilst the constants C, [epsilon] are arbitrary. This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the q's) executes a simple vibration of period 2[pi]/[sigma]. The amplitudes of oscillation of the various particles have definite ratios to one another, and the phases are in agreement, the absolute amplitude (depending on C) and the phase-constant ([epsilon]) being alone arbitrary. A vibration of this character is called a _normal mode_ of vibration of the system; the number n of such modes is equal to that of the degrees of freedom possessed by the system. These statements require some modification when two or more of the roots of the equation (6) are equal. In the case of a multiple root the minors of [Delta]([sigma]²) all vanish, and the basis for the determination of the quantities a_r disappears. Two or more normal modes then become to some extent indeterminate, and elliptic vibrations of the individual particles are possible. An example is furnished by the spherical pendulum (§ 13).

As an example of the method of determination of the normal modes we may take the "double pendulum." A mass M hangs from a fixed point by a string of length a, and a second mass m hangs from M by a string of length b. For simplicity we will suppose that the motion is confined to one vertical plane. If [theta], [phi] be the inclinations of the two strings to the vertical, we have, approximately,

2T = Ma²[.theta]² + m(a[.theta] + b[.psi])² \ (9) 2V = Mga[theta]² + mg(a[theta]² + b[psi]²). /

The equations (3) take the forms

a[:theta] + [mu]b[:phi] + g[theta] = 0, \ (10) a[:theta] + b[:phi] + g[phi] = 0. /

where [mu] = m/(M + m). Hence

([sigma]² - g/a)a[theta] + [mu][sigma]²b[phi] = 0, \ (11) [sigma]²a[theta] + ([sigma]² - g/b)b[phi] = 0. /

The frequency equation is therefore

([sigma]² - g/a)([sigma]² - g/b) - [mu][sigma]^4 = 0. (12)

The roots of this quadratic in [sigma]² are easily seen to be real and positive. If M be large compared with m, [mu] is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the former root, M swings almost like the bob of a simple pendulum of length a, being comparatively uninfluenced by the presence of m, whilst m executes a "forced" vibration (§ 12) of the corresponding period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two values of [sigma]² can never be exactly equal, but they are approximately equal if a, b are nearly equal and [mu] is very small. A curious phenomenon is then to be observed; the motion of each particle, being made up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of "beats" in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals. If, on the other hand, M is small compared with m, [mu] is nearly equal to unity, and the roots of (12) are [sigma]² = g/(a + b) and [sigma]² = mg/M·(a + b)/ab, approximately. The former root makes [theta] = [phi], nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the second mode a[theta] + b[phi] = 0, nearly, so that m is approximately at rest. The oscillation of M then resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends and subject to a tension mg.

The motion of the system consequent on arbitrary initial conditions may be obtained by superposition of the n normal modes with suitable amplitudes and phases. We have then

q_r = [alpha]_r[theta] + [alpha]_r´[theta]´ + [alpha]_r´´[theta]´´ + ..., (13)

where

[theta] = C cos ([sigma]t + [epsilon]), [theta]´ = C´ cos ([sigma]´t + [epsilon]), [theta]´´ = C´´ cos([sigma]´´t + [epsilon]), ... (14)

provided [sigma]², [sigma]´², [sigma]´´², ... are the n roots of (6). The coefficients of [theta], [theta]´, [theta]´´, ... in (13) satisfy the _conjugate_ or _orthogonal_ relations

a11[alpha]1[alpha]1´ + a22[alpha]2[alpha]2´ + ... + a12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0, (15) c11[alpha]1[alpha]1´ + c22[alpha]2[alpha]2´ + ... + c12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0, (16)

provided the symbols [alpha]_r, [alpha]_r´ correspond to two distinct roots [sigma]², [sigma]´² of (6). To prove these relations, we replace the symbols A1, A2, ... A_n in (5) by [alpha]1, [alpha]2, ... [alpha]_n respectively, multiply the resulting equations by a´1, a´2, ... a´_n, in order, and add. The result, owing to its symmetry, must still hold if we interchange accented and unaccented Greek letters, and by comparison we deduce (15) and (16), provided [sigma]² and [sigma]´² are unequal. The actual determination of C, C´, C´´, ... and [epsilon], [epsilon]´, [epsilon]´´, ... in terms of the initial conditions is as follows. If we write

C cos [epsilon] = H, -C sin [epsilon] = K, (17)

we must have

[alpha]_rH + [alpha]_r´H´ + [alpha]_r´´H´´ + ... = [q_r]0, \ (18) [sigma][alpha]_rH + [sigma]´[alpha]_r´H´ + [sigma]´´[alpha]_r´´H´´ + ... = [[.q]_r]0, /

where the zero suffix indicates initial values. These equations can be at once solved for H, H´, H´´, ... and K, K´, K´´, ... by means of the orthogonal relations (15).

By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to sums of squares. The transformation is in fact effected by the assumption (13), in virtue of the relations (15) (16), and we may write

2T = a[.theta]² + a´[.theta]´² + a´´[.theta]´´² + ..., \ (19) 2V = c[theta]² + c´[theta]´² + c´´[theta]´´² + .... /

The new co-ordinates [theta], [theta]´, [theta]´´ ... are called the _normal_ co-ordinates of the system; in a normal mode of vibration one of these varies alone. The physical characteristics of a normal mode are that an impulse of a particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a particular normal type maintains a displacement of that type only. The normal modes are further distinguished by an important "stationary" property, as regards the frequency. If we imagine the system reduced by frictionless constraints to one degree of freedom, so that the co-ordinates [theta], [theta]´, [theta]´´, ... have prescribed ratios to one another, we have, from (19),

c[theta]² + c´[theta]´² = c´´[theta]´´² + ... [sigma]² = ---------------------------------------------, (20) a[theta]² + a´[theta]´² + a´´[theta]´´² + ...

This shows that the value of [sigma]² for the constrained mode is intermediate to the greatest and least of the values c/a, c´/a´, c´´/a´´, ... proper to the several normal modes. Also that if the constrained mode differs little from a normal mode of free vibration (e.g. if [theta]´, [theta]´´, ... are small compared with [theta]), the change in the frequency is of the second order. This property can often be utilized to estimate the frequency of the gravest normal mode of a system, by means of an assumed approximate type, when the exact determination would be difficult. It also appears that an estimate thus obtained is necessarily too high.

From another point of view it is easily recognized that the equations (5) are exactly those to which we are led in the ordinary process of finding the stationary values of the function

V (q1, q2, ... q_n) ------------------------, T (q1, q2, ... q_n)

where the denominator stands for the same homogeneous quadratic function of the q's that T is for the [.q]'s. It is easy to construct in this connexion a proof that the n values of [sigma]² are all real and positive.

The case of three degrees of freedom is instructive on account of the geometrical analogies. With a view to these we may write

2T= a[.x]² + b[.y]² + c[.z]² + 2f[.y][.z] + 2g[.z][.x] + 2h[.x][.y], \ (21) 2V = Ax² + By² + Cz² + 2Fyz + 2Gzx + 2Hxy. /

It is obvious that the ratio

V (x, y, z) ----------- (22) T (x, y, z)

must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive. Denoting this value by [sigma]1², we have

Ax1 + Hy1 + Gz1 = [sigma]1²(ax1 + hy1 + [dP]gz1), \ Hx1 + By1 + Fz1 = [sigma]1²(hx1 + by1 + fz1), > (23) Gx1 + Fy1 + Cz1 = [sigma]1²(gx1 + fy1 + cz1), /

provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z. Again, the expression (22) will also have a least value when the ratios x : y : z are subject to the condition

[dP]V [dP]V [dP]V x1 ----- + y1 ----- + z1 ----- = 0; (24) [dP]x [dP]y [dP]z

and if this be denoted by [sigma]2² we have a second system of equations similar to (23). The remaining value [sigma]2² is the value of (22) when x : y : z arc chosen so as to satisfy (24) and

[dP]V [dP]V [dP]V x2 ----- + y2 ----- + z2 ----- = 0 (25) [dP]x [dP]y [dP]z

The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y, z) = const., V(x, y, z) = const. If in (21) we imagine that x, y, z denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of [sigma] are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.

We proceed to the _forced vibrations_ of the system. The typical case is where the extraneous forces are of the simple-harmonic type cos ([sigma]t + [epsilon]); the most general law of variation with time can be derived from this by superposition, in virtue of Fourier's theorem. Analytically, it is convenient to put Q_r, equal to e^(i[sigma]^t) multiplied by a complex coefficient; owing to the linearity of the equations the factor e^(i[sigma]^t) will run through them all, and need not always be exhibited. For a system of one degree of freedom we have

a[:q] + cq = Q, (26)

and therefore on the present supposition as to the nature of Q

Q q = -------------. (27) c - [sigma]²a

This solution has been discussed to some extent in § 12, in connexion with the forced oscillations of a pendulum. We may note further that when [sigma] is small the displacement q has the "equilibrium value" Q/c, the same as would be produced by a steady force equal to the instantaneous value of the actual force, the inertia of the system being inoperative. On the other hand, when [sigma]² is great q tends to the value -Q/[sigma]²a, the same as if the potential energy were ignored. When there are n degrees of freedom we have from (3)

(c_(1r) - [sigma]² a_(2r)) q1 + (c²_r - [sigma]² a_(2r)) q2 + ... + (c_(nr) - [sigma]² a_(nr)) q_n = Qr, (28)

and therefore

[Delta]([sigma]²)·q_r = a_(1r)Q1 + a_(2r)Q2 + ... + a_(nr)Q_n, (29)

where a_(1r), a_(2r), ... a_(nr) are the minors of the rth row of the determinant (7). Every particle of the system executes in general a simple vibration of the imposed period 2[pi]/[sigma], and all the particles pass simultaneously through their equilibrium positions. The amplitude becomes very great when [sigma]² approximates to a root of (6), i.e. when the imposed period nearly coincides with one of the free periods. Since a_(rs) = a_(sr), the coefficient of Q_s in the expression for q_r is identical with that of Q_r in the expression for q_s. Various important "reciprocal theorems" formulated by H. Helmholtz and Lord Rayleigh are founded on this relation. Free vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the complete solution of the dynamical equations.

In practice the vibrations of a system are more or less affected by dissipative forces. In order to obtain at all events a qualitative representation of these it is usual to introduce into the equations frictional terms proportional to the velocities. Thus in the case of one degree of freedom we have, in place of (26),

a[:q] + b[.q] + cq = Q, (30)

where a, b, c are positive. The solution of this has been sufficiently discussed in § 12. In the case of multiple freedom, the equations of small motion when modified by the introduction of terms proportional to the velocities are of the type

d [dP]T [dP]V --- ---------- + B_(1r)[.q]1 + B_(2r)[.q]2 + ... + B_(nr)[.q]_n + ------- = Q_r (31) dt [dP][.q]_r [dP]q_r

If we put

b_(rs) = b_(sr) = ½[B_(rs) + B_(sr)], [beta]_(rs) = -[beta]_(sr) = ½[B_(rs) - B_(sr)], (32)

this may be written

d [dP]T [dP]F [dP]V --- --------- + ---------- + [beta]_(1r)[.q]1 + [beta]_(2r)[.q]2 + ... + [beta]_(nr)[.q]_r + ------- (33) dt [dP][.q]_r [dP][.q]_r [dP]q_r

provided

2F = b11[.q]1² + b22[.q]2² + ... + 2b12[.q]1[.q]2 + ... (34)

The terms due to F in (33) are such as would arise from frictional resistances proportional to the absolute velocities of the particles, or to mutual forces of resistance proportional to the relative velocities; they are therefore classed as _frictional_ or _dissipative_ forces. The terms affected with the coefficients [beta]_(rs) on the other hand are such as occur in "cyclic" systems with latent motion (DYNAMICS, § _Analytical_); they are called the _gyrostatic terms_. If we multiply (33) by [.q]_r and sum with respect to r from 1 to n, we obtain, in virtue of the relations [beta]_(rs) = -[beta]_(sr), [beta]_(rr) = 0, d ---(T + V) = 2F + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (35) dt

This shows that mechanical energy is lost at the rate 2F per unit time. The function F is therefore called by Lord Rayleigh the _dissipation function_.

If we omit the gyrostatic terms, and write q_r = C_re^([lambda]t), we find, for a free vibration,

[a_(1r)[lambda]² + b_(1r)[lambda] + c_(1r)] C1 + [a_(2r)[lambda]² + b_(2r)[lambda] + c_(2r)] C2 + ... + [a_(nr)[lambda]² + b_(nr)[lambda] + c_(nr)] C_n = 0. (36)

This leads to a determinantal equation in [lambda] whose 2n roots are either real and negative, or complex with negative real parts, on the present hypothesis that the functions T, V, F are all essentially positive. If we combine the solutions corresponding to a pair of conjugate complex roots, we obtain, in real form,

q_r = C[alpha]_re^(-t/[tau]) cos ([sigma]t + [epsilon] - [epsilon]_r), (37)

where [sigma], [tau], [alpha]_r, [epsilon]_r are determined by the constitution of the system, whilst C, [epsilon] are arbitrary, and independent of r. The n formulae of this type represent a normal mode of free vibration: the individual particles revolve as a rule in elliptic orbits which gradually contract according to the law indicated by the exponential factor. If the friction be relatively small, all the normal modes are of this character, and unless two or more values of [sigma] are nearly equal the elliptic orbits are very elongated. The effect of friction on the period is moreover of the second order.

In a forced vibration of e^(i[sigma]t) the variation of each co-ordinate is simple-harmonic, with the prescribed period, but there is a retardation of phase as compared with the force. If the friction be small the amplitude becomes relatively very great if the imposed period approximate to a free period. The validity of the "reciprocal theorems" of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional forces of the kind here considered.

The most important applications of the theory of vibrations are to the case of continuous systems such as strings, bars, membranes, plates, columns of air, where the number of degrees of freedom is infinite. The series of equations of the type (3) is then replaced by a single linear partial differential equation, or by a set of two or three such equations, according to the number of dependent variables. These variables represent the whole assemblage of generalized co-ordinates q_r; they are continuous functions of the independent variables x, y, z whose range of variation corresponds to that of the index r, and of t. For example, in a one-dimensional system such as a string or a bar, we have one dependent variable, and two independent variables x and t. To determine the free oscillations we assume a time factor e^(i[sigma]t); the equations then become linear differential equations between the dependent variables of the problem and the independent variables x, or x, y, or x, y, z as the case may be. If the range of the independent variable or variables is unlimited, the value of [sigma] is at our disposal, and the solution gives us the laws of wave-propagation (see WAVE). If, on the other hand, the body is finite, certain terminal conditions have to be satisfied. These limit the admissible values of [sigma], which are in general determined by a transcendental equation corresponding to the determinantal equation (6).

Numerous examples of this procedure, and of the corresponding treatment of forced oscillations, present themselves in theoretical acoustics. It must suffice here to consider the small oscillations of a chain hanging vertically from a fixed extremity. If x be measured upwards from the lower end, the horizontal component of the tension P at any point will be P[delta]y/[delta]x, approximately, if y denote the lateral displacement. Hence, forming the equation of motion of a mass-element, [rho][delta]x, we have

[rho][delta]x·[:y] = [delta]P·([dP]y/[dP]x). (38)

Neglecting the vertical acceleration we have P = g[rho]x, whence

[dP]²y [dP] / [dP]y \ ------ = g ----- ( x ----- ). (39) [dP]t² [dP]x \ [dP]x /

Assuming that y varies as e^(i[sigma]t) we have

[dP] / [dP]y \ ----- ( x ----- ) + ky = 0 (40) [dP]x \ [dP]x /

provided k = [sigma]²/g. The solution of (40) which is finite for x = 0 is readily obtained in the form of a series, thus

/ kx k²x² \ y = C ( 1 - -- + ---- - ... ) = CJ0(z), (41) \ 1² 1²2² /

in the notation of Bessel's functions, if z² = 4kx. Since y must vanish at the upper end (x = l), the admissible values of [sigma] are determined by

[sigma]² = gz²/4l, J0(z) = 0. (42)

The function J0(z) has been tabulated; its lower roots are given by

z/[pi]= .7655, 1.7571, 2.7546,...,

approximately, where the numbers tend to the form s - ¼. The frequency of the gravest mode is to that of a uniform bar in the ratio .9815 That this ratio should be less than unity agrees with the theory of "constrained types" already given. In the higher normal modes there are nodes or points of rest (y = 0); thus in the second mode there is a node at a distance .190l from the lower end.

AUTHORITIES.--For indications as to the earlier history of the subject see W. W. R. Ball, _Short Account of the History of Mathematics_; M. Cantor, _Geschichte der Mathematik_ (Leipzig, 1880 ... ); J. Cox, _Mechanics_ (Cambridge, 1904); E. Mach, _Die Mechanik in ihrer Entwickelung_ (4th ed., Leipzig, 1901; Eng. trans.). Of the classical treatises which have had a notable influence on the development of the subject, and which may still be consulted with advantage, we may note particularly, Sir I. Newton, _Philosophiae naturalis Principia Mathematica_ (1st ed., London, 1687); J. L. Lagrange, _Mécanique analytique_ (2nd ed., Paris, 1811-1815); P. S. Laplace, _Mécanique céleste_ (Paris, 1799-1825); A. F. Möbius, _Lehrbuch der Statik_ (Leipzig, 1837), and _Mechanik des Himmels_; L. Poinsot, _Éléments de statique_ (Paris, 1804), and _Théorie nouvelle de la rotation des corps_ (Paris, 1834).

Of the more recent general treatises we may mention Sir W. Thomson (Lord Kelvin) and P. G. Tait, _Natural Philosophy_ (2nd ed., Cambridge, 1879-1883); E. J. Routh, _Analytical Statics_ (2nd ed., Cambridge, 1896), _Dynamics of a Particle_ (Cambridge, 1898), _Rigid Dynamics_ (6th ed., Cambridge 1905); G. Minchin, _Statics_ (4th ed., Oxford, 1888); A. E. H. Love, _Theoretical Mechanics_ (2nd ed., Cambridge, 1909); A. G. Webster, _Dynamics of Particles_, &c. (1904); E. T. Whittaker, _Analytical Dynamics_ (Cambridge, 1904); L. Arnal, _Traitê de mécanique_ (1888-1898); P. Appell, _Mécanique rationelle_ (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); G. Kirchhoff, _Vorlesungen über Mechanik_ (Leipzig, 1896); H. Helmholtz, _Vorlesungen über theoretische Physik_, vol. i. (Leipzig, 1898); J. Somoff, _Theoretische Mechanik_ (Leipzig, 1878-1879).

The literature of graphical statics and its technical applications is very extensive. We may mention K. Culmann, _Graphische Statik_ (2nd ed., Zürich, 1895); A. Föppl, _Technische Mechanik_, vol. ii. (Leipzig, 1900); L. Henneberg, _Statik des starren Systems_ (Darmstadt, 1886); M. Lévy, _La statique graphique_ (2nd ed., Paris, 1886-1888); H. Müller-Breslau, _Graphische Statik_ (3rd ed., Berlin, 1901). Sir R. S. Ball's highly original investigations in kinematics and dynamics were published in collected form under the title _Theory of Screws_ (Cambridge, 1900).

Detailed accounts of the developments of the various branches of the subject from the beginning of the 19th century to the present time, with full bibliographical references, are given in the fourth volume (edited by Professor F. Klein) of the _Encyclopädie der mathematischen Wissenschaften_ (Leipzig). There is a French translation of this work. (See also DYNAMICS.) (H. Lb.)

II.--APPLIED MECHANICS[1]

§ 1. The practical application of mechanics may be divided into two classes, according as the assemblages of material objects to which they relate are intended to remain fixed or to move relatively to each other--the former class being comprehended under the term "Theory of Structures" and the latter under the term "Theory of Machines."