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在回顾并分析了用Newton力学、Lagrange和Hamilton力学建立连续介质波动力学方程的基础上，提出了在能量守恒框架下建立波动力学方程的新思路与方法。首先，回顾了用牛顿第二定律推导波动力学方程，同时回顾并分析了利用Hamilton变分原理，推导了在连续介质中的Lagrange方程、Hamilton正则方程，以及相应的波动力学方程；其次，在能量守恒的框架下，建立了连续介质的Lagrange方程、Hamilton正则方程和波动力学方程，并证明了其结果与利用经典力学推导的结果的一致性，特别地，澄清了用Hamilton变分原理建立保守系统下连续介质的Lagrange方程和Hamilton正则方程时在边界条件应用时的一些模糊认识。在能量守恒框架下建立一系列动力学方程，为我们在不涉及泛函求极值的变分原理等基础上刻画和表述复杂介质中波动现象的演化规律提供了另一种途径，也深入探讨了最小作用原理的物理本质。最后，在能量守恒的框架下给出了建立粘弹性介质中的波动力学微分方程的应用。
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关键词:
- 波动力学方程 /
- Hamilton变分原理 /
- Lagrange方程 /
- 能量守恒
Based on the analysis of establishing dynamic equations by using Newton mechanics, Lagrange and Hamilton mechanics, a new idea of establishing dynamic equations under the framework of energy conservation is proposed. Firstly, the use of Newton’s second law to derive wave equations of motion is introduced. Secondly, the Lagrange equation, Hamilton canonical equation, and the corresponding dynamical equations in a continuum are derived by using Hamilton’s variational principle. Thirdly, under the framework of energy conservation, the Lagrange equation, Hamilton canonical equation, and acoustic dynamics equation of continuum are established, and the results are proved to be consistent with those derived from classical mechanics. Some of fuzzy understandings when using Hamilton's variational principle to establish Lagrange’s equation and Hamilton’s canonical equation, are clarified. A series of dynamical equations established under the framework of energy conservation provides alternative way to characterize and represent the propagation characteristics of wave motions in various complex media without involving the variational principle of functional extremum. Finally, as an application example, the differential equation of dynamics in viscoelastic medium is given under the framework of energy conservation.-
Keywords:
- Dynamical equation /
- Hamilton variational principle /
- Lagrange equation /
- law of conservation of energy
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