On the existence of a third integral of motion

Abstract

The aim of this paper is to unify the work done until now from different points of view on a third integral of motion besides the energy and the angular momentum integrals, and to present a number of applica- tions and generalizations. In the Introduction, different methods for finding integrals of motion are dis- cussed. The third integral is explicitly calculated by means of von Zeipel's method. In the next section different definitions of integrable systems are given and a distinction between useful and nonuseful integrals is made. Poincare 5 nonexistence theorem is mentioned and its many exceptions are pointed out. Then a distinction of separable and nonseparable systems is made. It is seen that by introducing contact trans- formations many dynamical systems become separable. This explains the fact that in the case of two degrees of freedom the orbits are in general distorted Lissajous figures. The problem of the convergence of the third integral is the subject of the next section. Siegel gave a proof that in general the formal third integral does not converge. However, one can approximate any Hamiltonian, that is given as a polynomial or a series, by another Hamiltonian that possesses converging integrals, coinciding with the given Hamiltonian up to the terms of any degree. Numerical results provide ample evidence that the third integral exists in quite general potential fields, as in Schmidt's model of the Galaxy, in slightly elliptical clusters or galaxies, in the field of the oblate earth, in resonance cases where the unperturbed frequences have a rational ratio, and in potential fields that do not have a plane of symmetry. In the case of non-axially symmetric systems two new integrals are introduced. However, no integral corresponds to the angular momentum. In an asymmetric galaxy initially circular orbits after some time pass near the center of the galaxy. The third integral can be applied to the boundaries of the orbits and their hodographs, to the study of periodic orbits, to the dispersion of a group of stars, to the three-axial velocity ellipsoid, to the construction of models of the Galaxy, etc. Finally a practical method for searching for new integrals is indicated.