DYNAMICS,
ANALYTICAL
now fix the position of every particle of the system. For example, if by suitable forces the system he brought back to its initial configuration (so far as this is defined by qv g2,... qm), after performing any evolutions, the ignored co-ordinates %, x',... will not in general return to their original values. If in Lagrange’s equations § 2 (10) we reverse the sign of the timeelement dt, the equations are unaltered. The motion is therefore reversible ; that is to say, if as the system is passing through any configuration its velocities qx, q^,... qn be all reversed, it will (if the forces be the same in the same configuration) retrace its former path. But it is important to observe that the statement does not in general hold of a gyrostatic system ; the terms of (16), which are linear in qlt q2,... qm, change sign with dt, whilst the others do not. Hence the motion of a gyrostatic system is not reversible, unless indeed we reverse the cyclic motions as well as the velocities f/j, ... qm. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin. The conditions of equilibrium of a system with latent cyclic are obtained by putting (/1 = 0, g'2 = 0,... qm — ^ Kiaeto- motions statics. in (16); viz., they are 0K 0K (23) These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration [qx, ^... qm) to rest in the configuration {qx + bqx, q% + <5y2,... qm + bqrnh the work done by the forces must be equal to the increment of the kinetic energy. Hence Ql% + Q252'2+-=SK) • • • (24) which is equivalent to (23). The conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy K. This is important from a physical point of view, as showing how energy which is apparently potential may in its ultimate essence be kinetic. By means of the formula (18), which now reduce to ,_0K .,_0K .„_0K X-0K> X —’0/c'’ s„/> X —'ok' •••> • • (25) K may also be expressed as a homogeneous quadratic function of the cyclic velocities x, x, x, ■.. . Denoting it in this form by T0, we have 5(T0 + K) = 2SK = S(kx + k'x' + k"x"+...) . • (26) Performing the variations, and omitting the terms which cancel by (2) and (25), we find 0To_ _0K 0To_ _0K • (27) 0<7i dqi 022 022’ so that the formulae (23) become, 0__?To O = dTp • . (28) •; Cqf 02! ’ ^ A simple example is furnished by the top (§ 2). The cyclic co-ordinates being i/', <p, we find) K <iqiL«i»2 £ 2& = Ad2 2 = A sm '6* + C 2T0=A sin22^'2 + C(0-l-i/'cos ^)2,. . (29) whence we may verify that 0Tp/00 = - 0K/00 in accordance with (27). And the condition of equilibrium 0K_ _0Y (30) 02 02 ' ’
gives the condition of steady precession, § 2 (26). Stability of Equilibrium and of Steady Motion. § 6. The theory of the small oscillations of a conservative system about a configuration of equilibrium, originated by Lagrange, has been greatly developed by Thomson and Tait, Routh, an< s la bilit 11 y 4 L°r(itoRayleigh. of the matter ’ properly acoustics, The but details some reference may bebelong made to the important but rather difficult question of stability. If we neglect small quantities of the second order in the velocities and the displacements, Lagrange’s equations assume a linear form, and the deviation of any co-ordinate from its equilibrium value is expressed2 by a series of terms of the type Ce , the admissible values of X being given by an algebraic equation of degree n. In order that the expressions for the deviations may not admit of indefinite increase, the values of X must be purely imaginary (i.e., of the form icr), so that the exponentials become replaced by circular functions of t. It is found that this will be the case if, and only if, the potential energy V is a minimum in the configuration of equilibrium. Accordingly, it is usual to say that the necessary and sufficient condition for stability is that the equilibrium-value of Y should be a minimum. The validity of this inference has in recent times been contested. It is pointed out that since Lagrange’s approximate equations
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become less and less accurate as the deviation from the equilibrium configuration increases, it is a matter for examination how far rigorous conclusions as to the ultimate extent of the deviation can be based upon them. Some researches in this direction have been instituted by Liapounoff, Poincare, and others, but the question cannot yet be regarded as completely resolved. That the occurrence of a minimum value of V is a sufficient condition of stability was shown independently by Dirichlet. 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 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 2i> 221 •• • in differ from zero. Hence, if the system be started from rest in a configuration for which V is less than in the equilibrium configuration, this expression must still further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some new equilibrium configuration, or V will in the long run diminish indefinitely. This argument is due to Thomson and Tait. A little consideration will show that a good deal of the obscurity which attaches to the question arises from the want of a sufficiently precise definition of what is meant by “stability.” Kinetic The same difficulty is encountered, in an aggravated stabilitv form, when we pass to the question of stability of motion. The various definitions of stability which have been propounded by different writers are examined critically by Klein and Sommerfeld in their book on the theory of the top. Rejecting previous definitions, they base their criterion of stability on the character of the changes produced in the path of the system by small arbitrary disturbing impulses. If the undisturbed path be the limiting form of the disturbed path when the impulses are indefinitely diminished, it is said to be stable, but not otherwise. For instance, the vertical fall of a particle under gravity is reckoned as stable, although for a given impulsive disturbance, however small, the deviation of the particle’s position at any time t from the position which it would have occupied in the original motion increases indefinitely with t. Even this criterion, as the writers quoted themselves recognize, is not free from ambiguity unless the phrase “limiting form,” as applied to a path, be strictly defined. It appears, moreover, that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions. A special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates qx, q^,... qm all vanish (see § 5). This has been discussed by Routh, Thomson and Tait, and Poincare. These writers treat the question, by an extension of Lagrange’s method, as a problem of small oscillations. Whether we adopt the notion of stability which this implies, or take up the position of Klein and Sommerfeld, there is no difficulty in showing that stability is ensured if V-f K be a minimum as regards variations of qx, q2,...qm. The proof is the same as that of Dirichlet for the case of statical stability. It is easily seen that, in the notation of § 5, an equivalent condition is that Y - T0 should be a minimum for the same variations. We can illustrate this condition from the case of the top, where, in our previous notation, v cos 2)2 t p2 Y + K=My/i cos 2 + {y.~ (1) 2A sin 22 +2C‘ To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put y—v. We then find without difficulty that V + K is a minimum provided v12 ^ lAMg/i. The method of small oscillations gave us the2 condition r >4AMyA, and indicated instability in the cases j/ YlAMyA. The present criterion can also be applied to show that the steady precessional motions in which the axis has a constant inclination to the vertical are stable. The question remains, as before, whether it is essential for stability that V + K should be a minimum. It appears that from the point of view of the theory of small oscillations it is not essential, and that there may even be stability when Y + K is a maximum. The precise conditions, which are of a somewhat elaborate character, have been formulated by Routh. An important distinction has, however, been established by Thomson and Tait, and by Poincare, between what we may call ordinary or accidental stability (which is stability in the above sense) and permanent or secular stability, which means stability when regard is had to possible dissipative
