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

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.

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References in corpus (4)

• Contact Hamiltonian Mechanics
• Geometrothermodynamics of black holes
• Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics
• Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics