### Small Oscillations to any Degree of Approximation

E. J. Routh
1873 Proceedings of the London Mathematical Society
In discussing the small oscillations of a dynamical system, we are usually content to reject the squares of all small quantities. It is clear, however, that in many cases the terms of the higher orders may rise in importance, and may even alter the period of the principal oscillation. It is proposed to investigate an easy method of, determining the oscillation of a heavy body moving in any manner with one independent motion to any degree of approximation. This we shall do, first, for any small
more » ... rst, for any small oscillation in two dimensions; and, secondly, for an oscillation in three dimensions about a fixed point. Let P be the instantaneous centre of rotation at the time t t AP its path in space, A'P in the body. Then the motion may be constructed by making the curve A'P, fixed in the body, roll without sliding on the curve AP fixed in space. Let AP = A'P = 8. Let Q be the centre of gravity of the body GP = r. Let ^ be the angle GP makes with the vertical,. n the angle it makes with the normal, to the curves AP, A'P at P. Let 0 be the angle any straight ]ine fixed in the body makes with a straight line fixed in space, «c the radius of gyration of the body about the centre of gravity. Taking moments in the usual way about the instantaneous centre of rotation considered as a moving point, the equation of motion is £ ? + rz sin n (^) = gr sin *. The method of proceeding is as follows :-To solvo the equation, we must expand each coefficient by Taylor's theorem in powers of 0, which is to bo so chosen as to vanish in the position of equilibrium. To do this we require the successive differentials of the coefficients to any order expressed in terms of the initial values only of \p, n, and r. We find by inspection of the figure Ie = l -Z cos n dn _ /cos 11 __ 1 \ dd~Z\ r 7/' dr dd = z sin ft,