Invariant variational principle for Hamiltonian mechanics

TL;DR

This paper demonstrates that Hamiltonian equations of motion can be formulated using an invariant symplectic form, eliminating the need for a specific 1-form and ensuring a more fundamental geometric description.

Contribution

It introduces an invariant variational principle for Hamiltonian mechanics that is expressed directly in terms of the symplectic structure, avoiding non-unique or non-global 1-forms.

Findings

Hamiltonian equations can be formulated with an invariant symplectic form.

The formulation does not require choosing a specific 1-form γ.

This approach provides a more fundamental geometric perspective.

Abstract

It is shown that the action for Hamiltonian equations of motion can be brought into invariant symplectic form. In other words, it can be formulated directly in terms of the symplectic structure $\omega$ without any need to choose some 1-form $\gamma$, such that $\omega= d \gamma$, which is not unique and does not even generally exist in a global sense.