A generalized Hamiltonian formulation of the principle of virtual work

TL;DR

This paper extends the Hamiltonian formulation of the principle of virtual work, allowing for decomposition of Lie brackets and generalization of the Hamilton-Jacobi equation within symplectic mechanics.

Contribution

It introduces a generalized Hamiltonian formulation derived from a variational principle, enabling new decompositions and generalizations in symplectic and contact geometry.

Findings

Decomposition of Lie brackets into Poisson brackets terms.

Generalization of the Hamilton-Jacobi equation to nonlinear PDEs.

Extension of variational principles to broader Hamiltonian frameworks.

Abstract

The authors previous derivation of a variational principle from the total work functional, as a generalization of the first variation of an action functional, is extended by deriving a corresponding generalization of the Hamiltonian formulation of that action functional. Some consequences of it are that one can decompose the Lie brackets of arbitrary vector fields on symplectic mechanics into a sum of terms that involve the Poisson brackets of the functions that appear in the normal form of the Pfaffian that is symplectic-dual to the vector field and that one can also generalize the Hamilton-Jacobi equation to a system of nonlinear first-order partial differential equations for the contact field that one uses in order to obtain them.