Geometrical structures on the cotangent bundle

TL;DR

This paper explores the geometric structures on the cotangent bundle, focusing on adapted tangent structures, regular vector fields, and their relation to Hamiltonian and Lagrangian formalisms in differential geometry.

Contribution

It establishes a connection between semi-Hamiltonian vector fields on the cotangent bundle and semisprays on the tangent bundle via nonlinear connections and Legendre transformation.

Findings

Dynamical covariant derivative fixes a nonlinear connection for a given $\

Semi-Hamiltonian vector fields correspond to semisprays under Legendre transformation.

The canonical nonlinear connection on $TM$ is characterized by the regular Lagrangian.

Abstract

In this paper we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on $T^{*}M$ fix a nonlinear connection for a given $\mathcal{J}$-regular vector field. Using the Legendre transformation induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector field on $T^{*}M$ corresponds to a semispray on $TM$ if and only if the nonlinear connection on $TM$ is just the canonical nonlinear connection induced by the regular Lagrangian.