Canonical and canonoid transformations for Hamiltonian systems on (co)symplectic and (co)contact manifolds

TL;DR

This paper explores the geometric structures of Hamiltonian systems on (co)symplectic and (co)contact manifolds, defining and characterizing canonoid transformations and their associated constants of motion.

Contribution

It introduces a unified geometric framework for canonoid transformations across various Hamiltonian systems, extending their definitions and properties.

Findings

Explicit local characterizations of canonoid transformations

Existence of constants of motion linked to canonoid transformations

Unified treatment across symplectic, cosymplectic, contact, and cocontact geometries

Abstract

In this paper we present canonical and canonoid transformations considered as global geometrical objects for Hamiltonian systems. Under the mathematical formalisms of symplectic, cosymplectic, contact and cocontact geometry, the canonoid transformations are defined for (co)symplectic, (co)contact Hamiltonian systems, respectively. The local characterizations of these transformations is derived explicitly and it is demonstrated that for a given canonoid transformation there exist constants of motion associated with it