基于能量守恒原理建立分析力学动力学方程的方法论
A methodology for formulating dynamical equations in analytical mechanics based on the principle of energy conservation
AI总结：
本文提出以能量守恒原理统一推导分析力学动力学方程的方法论，阐明Lagrange、Hamilton等方程及相关守恒律的能量本质，并用于耗散、多场耦合系统及推导Cauchy第一运动定律。
摘要：
在本工作中，提出了一种在能量守恒原理框架下建立力学中一般动力学方程的方法论。研究表明，Lagrange方程、Hamilton正则方程和Hamilton-Jacobi方程均基于能量守恒原理，并通过一个简单的能量守恒方程来建立，即动能与势能随时间的变化率等于外力所做功随时间的变化率；而D'Alembert原理是能量守恒定律的一种特殊情形，其中采用虚位移（“冻结”时间）或虚位移（“冻结”广义坐标）。本文论证指出，所有用于表征系统动力学行为的表述都可由能量守恒原理导出，并且从广义上讲，能量守恒原理是构建力学的根本指导。所提出的方法论为处理一般力学中的动力学问题提供了有效途径，包括耗散连续介质系统，尤其适用于具有多物理场相互作用与耦合的系统。文中指出，与经典分析力学、特别是现有Hamilton力学相反，Hamilton变分原理、Lagrange方程和Newton第二运动定律的物理本质，包括它们的推论如动量守恒和角动量守恒，都是能量守恒定律的结果。此外，我们提出的方法论更易于理解，具有清晰的物理意义，并可用于解释现有的力学原理或定理。最后，作为一个应用实例，该方法论被应用于流体力学，以推导Cauchy第一运动定律。
英文摘要：
In this work, a methodology is proposed for formulating general dynamical equations in mechanics under the umbrella of the principle of energy conservation. It is shown that Lagrange's equation, Hamilton's canonical equations, and Hamilton-Jacobi's equation are all formulated based on the principle of energy conservation with a simple energy conservation equation, i.e., the rate of kinetic and potential energy with time is equal to the rate of work with time done by external forces; while D'Alembert's principle is a special case of the law of the conservation of energy, with either the virtual displacements ('frozen' time) or the virtual displacement ('frozen' generalized coordinates). It is argued that all of the formulations for characterizing the dynamical behaviors of a system can be derived from the principle of energy conservation, and the principle of energy conservation is an underlying guide for constructing mechanics in a broad sense. The proposed methodology provides an efficient way to tackle the dynamical problems in general mechanics, including dissipation continuum systems, especially for those with multi-physical field interactions and couplings. It is pointed out that, on the contrary to the classical analytical mechanics, especially to existing Hamiltonian mechanics, the physics essences of Hamilton's variational principle, Lagrange's equation, and the Newtonian second law of motion, including their derivatives such as momentum and angular momentum conservations, are the consequences of the law of conservation of energy. In addition, our proposed methodology is easier to understand with clear physical meanings and can be used for explaining the existing mechanical principles or theorems. Finally, as an application example, the methodology is applied in fluid mechanics to derive Cauchy's first law of motion.
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