为非完整约束系统构建通用变分原理，直接导出正确动力学。
Variational approach to nonholonomic and inequality-constrained mechanics
- 基于量子路径积分的类比，构造可极小化的标量作用量
- 数值优化验证了其对典型系统的正确性，无需解方程
- 为约束系统提供新分析与计算工具，适合理论物理研究者
变分原理在经典力学中占据核心地位，能简洁表述动力学并直接获得守恒量。虽然完整系统已有成熟的动作量形式，但受非积分速度约束或位置不等式约束的非完整系统长期缺乏统一的作用量极小化处理。本文基于量子施温格-克尔迪什作用量形式的经典极限（由加利重新发现），构建了一个显式的通用作用量，通过极小化该标量作用量可恢复拉格朗日-达朗贝尔方程的动力学。我们通过直接数值优化新作用量，在典型例子中验证了该方法的有效性，绕过了运动方程的求解。本框架拓展了变分力学的应用范围，为约束系统提供了新的分析与计算工具。
原文摘要 · Abstract (English)
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.
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