达朗贝尔 - 拉格朗日原理与拉格朗日方程
The D'Alembert-Lagrange Principle and Lagrange Equations
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中文总结 AI 辅助
该文为教育目的，对达朗贝尔 - 拉格朗日原理严格数学表述并推导拉格朗日方程，通过建立扩展相空间几何框架，利用变分导数协变性，从动力学一般方程导出完整系统第二类拉格朗日方程。
中文摘要 AI 辅助
本文是一篇方法指南，对达朗贝尔 - 拉格朗日原理进行了严格数学表述，并推导了受理想约束的机械系统的拉格朗日方程。为教育目的而作，为扩展相空间建立了清晰几何框架，独立于具体解析表示定义了虚位移空间和约束反力。核心是从动力学一般方程过渡到完整系统的第二类拉格朗日方程，通过考虑变分导数在构型流形嵌入映射下的协变性自然优雅地导出拉格朗日方程。
英文摘要
This expository article serves as a methodical guide, presenting a rigorous mathematical formulation of the D'Alembert-Lagrange principle and the derivation of the Lagrange equations for mechanical systems subject to ideal constraints. Designed for educational purposes, the paper establishes a clear geometric framework for the extended phase space, defining the space of virtual displacements and constraint reactions independently of their specific analytical representations. A central focus of this instructional text is the transition from the general equation of dynamics to the Lagrange equations of the second kind for holonomic systems. Specifically, we demonstrate that the Lagrange equations can be naturally and elegantly derived from considerations of the covariance of the variational derivative under the embedding mapping of the configuration manifold.
