An Underlying Geometrical Manifold for Hamiltonian Mechanics

TL;DR

This paper establishes a geometric framework for Hamiltonian mechanics using an underlying manifold with a conformal metric, linking it to traditional Hamilton-Lagrange mechanics and analyzing orbit variations and instabilities.

Contribution

It introduces a geometric representation of Hamiltonian mechanics with a conformal metric, providing a method to relate potentials and analyze orbit variations.

Findings

A conformal metric can be expanded in terms of the potential functions.

Orbit variations in Hamilton-Lagrange mechanics correspond to geodesic variations in the geometric picture.

The geometric framework offers insights into the manifestation of dynamical instabilities.

Abstract

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture) that can be put into correspondence with the usual Hamilton-Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamilton-Lagrange picture, defined on the Hamilton-Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton-Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton-Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the original Hamiltonian motion.