Variational approach to nonholonomic and inequality-constrained mechanics

TL;DR

This paper develops a new variational principle for non-holonomic and inequality-constrained mechanical systems, enabling direct extremization of an explicit action to recover correct dynamics.

Contribution

It introduces a general action formulation for non-holonomic systems inspired by quantum field theory, validated through numerical optimization and applicable to constrained mechanics.

Findings

Successfully recovers Lagrange-d'Alembert equations via extremization

Validates the approach on canonical examples using numerical methods

Extends variational mechanics to a broader class of constrained systems

Abstract

Variational principles play a central role in classical mechanics, providing compact formulations of dynamics and direct access to conserved quantities. While holonomic systems admit well-known action formulations, non-holonomic systems -- subject to non-integrable velocity constraints or position inequality constraints -- have long resisted a general extremized action treatment. In this work, we construct an explicit and general action for non-holonomic motion, motivated by the classical limit of the quantum Schwinger-Keldysh action formalism, rediscovered by Galley. Our formulation recovers the correct dynamics of the Lagrange-d'Alembert equations via extremization of a scalar action. We validate the approach on canonical examples using direct numerical optimization of the novel action, bypassing equations of motion. Our framework extends the reach of variational mechanics and offers new analytical and computational tools for constrained systems.