On the Ideal Resonance Problem--II

Abstract

A number of problems in celestial mechanics, as well as other branches of applied mathematics, possess an ideal resonance structure. That is, the Hamiltonian F , of the dynamical system, may be written in the form |$-F\,=\,B(x)\,+\,2\mu A(x)\,\text{sin}^{2}\,y$| , where x and y are the canonically conjugate momentum and coordinate variables respectively and μ is a small constant parameter. Resonance is associated with the vanishing of the first derivative of B for some value of x . It is well known that the phase-plane of such a system is separated into libration and circulation regions by the limiting orbit. The author has previously published a paper which presented a new method of solution for the region of libration. In the present publication the general nature of the higher order solution is discussed. The motivation for developing the new solution, characterized by a transformation due to Poincaré, is explained. It is found that, to any power of μ½ , an asymptotic solution can be constructed, free from singularities and mixed secular terms. General expressions are given which demonstrate that the solution can always be expressed in terms of a small number of elliptic functions and fundamental integrals, defined in the text.