Hessian-information geometric formulation of Hamiltonian systems and generalized Toda's dual transform

TL;DR

This paper develops a geometric framework for a class of Hamiltonian systems with convex energy functions, introduces a generalized Toda dual transform, and explores applications to electric circuit models.

Contribution

It presents a Hessian-information geometric formulation for convex Hamiltonian systems and introduces a novel generalized Toda's dual transform.

Findings

Relation between Toda's dual transform and Legendre transform.

Geometric formulation applicable to electric circuit models.

Extension of the framework to dissipative systems not covered.

Abstract

In this paper a class of classical Hamiltonian systems is geometrically formulated. This class is such that a Hamiltonian can be written as the sum of a kinetic energy function and a potential energy function. In addition, these energy functions are assumed strictly convex. For this class of Hamiltonian systems Hessian and information geometric formulation is given. With this formulation, a generalized Toda's dual transform is proposed, where his original transform was used in deriving his integrable lattice system. Then a relation between the generalized Toda's dual transform and the Legendre transform of a class of potential energy functions is shown. As an extension of this formulation, dissipation-less electric circuit models are also discussed in the geometric viewpoint above.