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 the Lagrange-d'Alembert equations with nonlinear nonholonomic constraints. It introduces a universal variational principle with an auxiliary field $e(t)$ and derives a Hamiltonian formulation on a dimension $4n+2$ phase space that reproduces the LDA dynamics, including dissipative forces. The construction uses Ostrogradsky reduction and Dirac constraint analysis, yielding a Hamiltonian with primary constraints and a gauge degree of freedom in $e(t)$. The Appendix's Chaplygin sleigh analysis confirms the approach by showing correct infinite-friction behavior for LDA and highlighting the shortcomings of vakonomic formulations.

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.