论文标题
动作原理的几何和物理解释
Geometric and physical interpretation of the action principle
论文作者
论文摘要
我们对经典拉格朗日粒子力学中固定作用的原理给出了几何解释。简而言之，沿路径及其变化的动作的差异有效地``计数''可能的演变``经过''区域所包围的区域。如果路径对应于可能的演化，则所有相邻的演变都将平行，使它们与路径及其变化所包围的区域相切，从而产生固定的作用。这种治疗方法对哈密顿和拉格朗日力学的几何形状提供了完整的物理描述，这些几何形状建立在三个假设上：确定性和可逆进化，自由度的独立性以及运动学和动力学之间的等效性。这三个假设与固定行动原则之间的逻辑等效性导致了更加清洁的概念理解。
We give a geometrical interpretation for the principle of stationary action in classical Lagrangian particle mechanics. In a nutshell, the difference of the action along a path and its variation effectively ``counts'' the possible evolutions that ``go through'' the area enclosed. If the path corresponds to a possible evolution, all neighbouring evolutions will be parallel, making them tangent to the area enclosed by the path and its variation, thus yielding a stationary action. This treatment gives a full physical account of the geometry of both Hamiltonian and Lagrangian mechanics which is founded on three assumptions: determinism and reversible evolution, independence of the degrees of freedom and equivalence between kinematics and dynamics. The logical equivalence between the three assumptions and the principle of stationary action leads to a much cleaner conceptual understanding.
