Variational problem and Hamiltonian formulation of the Lagrange-d'Alembert equations with nonlinear nonholonomic constraints

TL;DR

This paper develops a variational and Hamiltonian framework for systems with nonlinear nonholonomic constraints, extending classical mechanics formulations to include dissipative forces and providing a unified approach.

Contribution

It introduces a novel variational principle with local symmetry that yields Hamiltonian equations for constrained systems, including dissipative forces.

Findings

Derived Hamiltonian formulation in a high-dimensional phase space.

Applied framework to nonlinear nonholonomic constraints.

Extended to systems with dissipative (frictional) forces.

Abstract

Any given system of ordinary differential equations in $n$-dimensional configuration space can be obtained from a peculiar variational problem with one local symmetry. The obtained action functional leads to the Hamiltonian formulation in $(4n+2)$-dimensional phase space. As concrete examples, we discuss the cases of Lagrange-d'Alembert equations with nonlinear nonholonomic constraints, as well as the equations of motion with dissipative (frictional) forces.