A new approach to the librational solution in the Ideal Resonance Problem

Abstract

The librational motion of the Ideal Resonance Problem (Garfinkel, 1966, Jupp, 1969) is treated through an initialnon-canonical transformation which, however, leaves the equations of motion in a ‘quasi-canonical’ form, with Hamiltonian expressed in standard trigonometric functions amenable to traditional averaging techniques. The perturbed solutions, similarly expressed intrigonometric near-identity transformations, and their frequencies can be found to arbitrary order, with the elliptic integrals expected of the system introduced only in a final explicit quadrature for a Kepler-type equation in the angular variable. The specific transformations and resulting equations of motion are introduced, and explicit solutions for the original variables are found to second order, with ‘mean motion’ accurate to fifth. Limitation of the present solution to the librational region, the extension of that solution to higher order, and observations on the form of the associated Hamiltonian are also discussed.