从经典标度变换导出的不变关系
Invariant relationships deriving from classical scaling transformations
摘要：
因为欧拉-拉格朗日方程的标度对称性通常不是作用量的变分对称性，所以它们不会导致守恒律。相反，诺特定理的扩展将运动方程简化为证明有用的演化定律，即使变换不是运动方程的广义对称。在缩放的情况下，对称性导致缩放的演化规律，一阶方程的尺度不变量，线性相关的运动学和动力学的自由度。这种标度演化规律在动态系统和静态系统中都有体现。应用于动力学中心力系统，标度演化方程导致广义维里定律，线性连接的动能和势能。应用于正压流体静力球，标度演化方程线性连接的重力和内能密度。这意味着众所周知的多面体的性质，描述退化的恒星和化学均匀的非退化恒星核心。
英文摘要：
Because scaling symmetries of the Euler-Lagrange equations are generally not variational symmetries of the action, they do not lead to conservation laws. Instead, an extension of Noether's theorem reduces the equations of motion to evolutionary laws that prove useful, even if the transformations are not generalized symmetries of the equations of motion. In the case of scaling, symmetry leads to a scaling evolutionary law, a first-order equation in terms of scale invariants, linearly relating kinematic and dynamic degrees of freedom. This scaling evolutionary law appears in dynamical and in static systems. Applied to dynamical central-force systems, the scaling evolutionary equation leads to generalized virial laws, which linearly connect the kinetic and potential energies. Applied to barotropic hydrostatic spheres, the scaling evolutionary equation linearly connects the gravitational and internal energy densities. This implies well-known properties of polytropes, describing degenerate stars and chemically homogeneous non-degenerate stellar cores.
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