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.

翻译：变分原理在经典力学中占据核心地位，为动力学提供了简洁的表述形式，并可直接导出守恒量。虽然完整约束系统存在经典的作用量表述，但受限于不可积速度约束或位置不等式约束的非完整系统，长期以来一直缺乏普适的极值作用量处理方法。本研究基于Galley重新发现的量子Schwinger-Keldysh作用量形式的经典极限，构建了适用于非完整运动的显式普适作用量。我们的表述通过标量作用量的极值化，恢复了Lagrange-d'Alembert方程的正确动力学。我们通过对新型作用量进行直接数值优化（无需借助运动方程），在典型示例上验证了该方法的有效性。该框架拓展了变分力学的适用范围，并为约束系统提供了新的解析与计算工具。