一般Lagrange系统的Jacobi度量
On the Jacobi metric for a general Lagrangian system
摘要：
给出了一般拉格朗日系统的Jacobi度量的显式表达式，它是系统动能平方根的级数展开式，相应的测地线用适当的非线性连接和曲率表示.在低动能的限制下，任何拉格朗日系统的运动轨迹都可以很好地近似于与能量相关的兰德斯度规的测地线，或者等价地，可以近似于受类电磁和类引力力场作用的代表点的位形空间中的路径。对于更高的动能值的运动轨迹，而不是一个一般的能量依赖芬斯勒度量的测地线，相应的配置空间中的路径的一个代表点受到一个层次的潜在无限数量的协变力场，概括了电磁和引力的。一些一般性的影响，这些研究结果进行了讨论。
英文摘要：
An explicit expression for the Jacobi metric for a general Lagrangian system is obtained as a series expansion in the square root of the kinetic energy of the system and the corresponding geodesics are described in terms of an appropriate non-linear connection and the associated curvature. In the limit of low kinetic energies the trajectories of motion of any Lagrangian system are very well approximated by the geodesics of an energy dependent Randers metric or, equivalently, by the paths in configuration space of a representative point subject to electromagnetic- and gravitational-like force fields. For higher kinetic energy values the trajectories of motion are instead the geodesics of a general energy dependent Finsler metric, corresponding also to the paths in configuration space of a representative point subject to a hierarchy of a potentially infinite number of covariant force fields that generalise the electromagnetic and gravitational ones. Some general implications of these findings are discussed.
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