The energy of a perturbed heavy symmetric top in periodic motion (Q1111785)

It is known that the complete solution of the equations of motion of a symmetric top, which is acted upon only by gravity, leads to a great number of conclusions about the qualitative characteristics of the top's behaviour. The general solution includes elliptic functions which, though they are quite interesting for extensive mathematical discussions are not always the best way to answer all the particular questions about the top's behaviour. For example, in order to check the symmetry of some periodic motions it is preferable to apply the appropriate transformation directly into the equations of motion instead of investigating the analytical expression of the general solution. This indirect way of reaching conclusions about the qualitative characteristics of motion by using a technique which is not based on the general solution, is applied throughout this paper. More precisely, we show, by means of the canonical equations of motion, that the energy of a heavy symmetric top in periodic motion remains unchanged under small perturbations of the initial conditions. This is true for any direction in the phase space, except the one corresponding to the eigenvector of the second zero characteristic exponent of the matrix of first variations. This theoretical conclusion enables one to find families of isoenergetic periodic motions and then to choose for practical reasons those motions which have the ``smallest'' period.