论文标题
奇异的Lagrangians和Dirac-古典力学中的伯格曼算法
Singular Lagrangians and the Dirac--Bergmann Algorithm in Classical Mechanics
论文作者
论文摘要
经典力学的教科书治疗通常假定Lagrangian是非语言的。也就是说，拉格朗日第二个衍生物相对于速度的矩阵是可逆的。该假设确保（i）拉格朗日的方程可以作为坐标和速度的函数的加速度来求解，并且（ii）可以将共轭力动量的定义倒置，以作为坐标和力量的函数的速度倒置。但是，这个假设是不必要的限制性的 - 有一些有趣的经典动力系统具有奇异的拉格朗日人。 Dirac和Bergmann在1950年代开发了用于分析此类系统的算法。经过对狄拉克 - 伯格曼算法的简要回顾后，由熟悉的组件构成了几个实例：通过质量弹簧，杆，绳索，绳索和皮带轮连接的点质量。该算法还用于开发具有全能约束的系统的初始价值公式。
Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular. That is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption insures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definition of the conjugate momenta can be inverted for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive -- there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950's. After a brief review of the Dirac--Bergmann algorithm, several physical examples are constructed from familiar components: point masses connected by massless springs, rods, cords and pulleys. The algorithm is also used to develop an initial value formulation of systems with holonomic constraints.
