Canonical Transformation of Potential Model Hamiltonian Mechanics to Geometrical Form I

TL;DR

This paper demonstrates a symplectic geometric approach to transform potential Hamiltonian systems into a geometric form on manifolds, enabling stability analysis via geodesic deviation and providing an algorithm for the transformation.

Contribution

It introduces a method to convert potential Hamiltonians into a geometric form using canonical transformations, with a convergence proof for the algorithm in one dimension.

Findings

Existence of a canonical transformation to geometric form

Algorithm for finding the generating function

Relation between stability in both representations

Abstract

Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the corresponding motions are along geodesic curves. The advantage of this representation is that it admits the computation of geometric deviation as a test for local stability, shown in previous studies to be a very effective criterion for the stability of the orbits generated by the potential model Hamiltonian. We describe here an algorithm for finding the generating function for the canonical transformation and describe some of the properties of this mapping under local diffeomorphisms. We give a convergence proof for this algorithm for the one-dimensional case, and provide a precise geometric formulation of geodesic deviation which relates the stability of the motion in the geometric form to that of the Hamiltonian standard form. We discuss the relation of bounded domains in the two representations for which Morse theory would be applicable. Numerical computations for some interesting examples will be presented in forthcoming papers.