The ubiquity of the symplectic hamiltonian equations in mechanics

TL;DR

This paper demonstrates that Hamiltonian equations, derived from geometric principles, are widely applicable across various mechanical systems, including nonholonomic and algebroid-based systems, through a unifying symplectic formalism.

Contribution

It introduces a Hamiltonian formalism applicable to diverse mechanical systems using geometric structures, especially algebroids, and provides a symplectic realization for these systems.

Findings

Hamiltonian formalism extends to nonholonomic and algebroid systems.

A symplectic realization for systems on algebroids is achieved.

The geometric approach unifies various mechanical systems under Hamiltonian theory.

Abstract

In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry characterizing the different systems. In particular, we obtain that the class of the so-called algebroids covers a great variety of mechanical systems. Finally, as the main result, a hamiltonian symplectic realization of systems defined on algebroids is obtained.