Horace Lamb

“ We are thus led to Lagrange’s equation of motion for a conservative system. It appears that the formula (12) is a convenient as well as a compact embodiment of the whole of ordinary dynamics.
The modification of the Hamiltonian principle appropriate to the case of cyclic systems has been given by J. Larmor. ”

“ 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 all vanish (see § 5) . This has been discussed by Routh, Lord Kelvin and Tait, and Poincaré. 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 be a minimum as regards variations of . ”

“ Continuous Motion of a System.
We may proceed to the continuous motion of a system. The equations of motion of any particle of the system are of the formLagrange’s equations.
Now let be the co-ordinates of in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation
Lagrange’s investigation consists in the transformation of (2) into an equation involving the independent variations . ”

“ A number of definitions which have been propounded by different writers are examined by F. Klein and A. Sommerfeld in their work Über die Theorie des Kreisels (1897–1903) . 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. ”

“ These methods, in addition to the unity and breadth which they have introduced into the treatment of pure dynamics, have a peculiar interest in relation to modern physical speculation, which finds itself confronted in various directions with the problem of explaining on dynamical principles the properties of systems whose ultimate mechanism can at present only be vaguely conjectured. ”

“ 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 from the position which it would have occupied in the original motion increases indefinitely with . ”

“ It is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. ”

“ As an example, let be the co-ordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of . If be the angular co-ordinate of the solid, we find without difficulty
where is the moment of inertia of the solid. The equations of motion, viz. ”

“ But if the amplitude of that circular vibration which agrees in sense with the rotation will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. ”

“ The remaining dynamical equations, equal in number to the co-ordinates , yield expressions for the forces which must be applied in order to maintain the velocities constant; they need not be written down. ”

“ The proof of (12) is simple; we have
The integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d’Alembert’s principle.
The fact that in (12) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. Thus, to deduce Lagrange’s equations, we have
The integrated terms vanish at both limits ”