Contact geometric mechanics: the Tulczyjew triples

TL;DR

This paper extends the Tulczyjew triple framework to contact geometry, enabling a geometric formulation of Hamiltonian and Lagrangian mechanics on contact manifolds with new structures and examples.

Contribution

It introduces a generalized Tulczyjew triple for contact manifolds, including contact Hamiltonians, Lagrangians, and a contact Legendre transformation, broadening geometric mechanics tools.

Findings

Developed a contact analog of the Tulczyjew triple.

Defined contact Hamiltonians and Lagrangians as sections of line bundles.

Provided explicit examples illustrating the contact formalism.

Abstract

We propose a generalization of the classical Tulczyjew triple as a geometric tool in Hamiltonian and Lagrangian formalisms which serves for contact manifolds. The r\^ole of the canonical symplectic structures on cotangent bundles in Tulczyjew's case is played by the canonical contact structures on the bundles $J^1L$ of first jets of sections of line bundles $L\to M$. Contact Hamiltonians and contact Lagrangians are understood as sections of certain line bundles, and they determine (generally implicit) dynamics on the contact phase space $J^1L$. We also study a contact analog of the Legendre map and the Legendre transformation of generating objects in both contact formalisms. Several explicit examples are offered.