论经典力学中的对称性与守恒律
On Symmetry and Conserved Quantities in Classical Mechanics
摘要：
本文阐述了在经典力学的拉氏框架和哈密顿框架下，连续对称性与守恒量即Noether "第一定理“之间的关系。这说明了力学的一个伟大主题：利用对称性来减少处理问题所需的变量数量。我强调的是，对于这两个框架，定理是由循环坐标的想法，和汉密尔顿定理是更强大的。拉格朗日定理的主要“成分”，除了循环坐标，是整流的向量场所提供的局部存在性和唯一性的解决方案，以常微分方程。对于哈密顿定理，主要的额外成分是泊松括号的不对称性，以及向量场产生正则变换当且仅当它是哈密顿的。
英文摘要：
This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's ``first theorem'', in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem's main ``ingredient'', apart from cyclic coordinates, is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian.
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