标题： 广义李对称性和几乎正则拉格朗日量：对称性与动力学之间的联系 显示英文标题

标题： Generalized Lie Symmetries and Almost Regular Lagrangians: A Link Between Symmetry and Dynamics

摘要： 研究了几乎正则拉格朗日量的广义李对称性，以及它们对动力系统演化的影响。 发现如果作用量具有广义李对称性，则拉格朗日量必定是奇异的；反之则不成立，我们通过一个具体例子加以说明。 还发现作用量的广义李对称性是运动欧拉-拉格朗日方程的广义李对称性的李子群。 反之再次不成立，存在一些系统，其运动的欧拉-拉格朗日方程具有广义李对称性，而作用量没有，我们再次通过一个具体例子加以说明。 最重要的是，证明了作用量的每个广义李对称性都会为动力系统的演化贡献一个任意函数。 这种对称性的数量给出了从拉格朗日相空间中任何允许的初始数据集出发的曲线族维度的一个下限。 此外，如果在应用拉格朗日约束算法时引入了二阶或更高阶的拉格朗日约束，这些额外的约束不可能是由于作用量的广义李对称性引起的。 显示英文摘要

摘要： The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on the evolution of dynamical systems is determined. It is found that if the action has a generalized Lie symmetry, then the Lagrangian is necessarily singular; the converse is not true, as we show with a specific example. It is also found that the generalized Lie symmetry of the action is a Lie subgroup of the generalized Lie symmetry of the Euler-Lagrange equations of motion. The converse is once again not true, and there are systems for which the Euler-Lagrange equations of motion have a generalized Lie symmetry while the action does not, as we once again show through a specific example. Most importantly, it is shown that each generalized Lie symmetry of the action contributes one arbitrary function to the evolution of the dynamical system. The number of such symmetries gives a lower bound to the dimensionality of the family of curves emanating from any set of allowed initial data in the Lagrangian phase space. Moreover, if second- or higher-order Lagrangian constraints are introduced during the application of the Lagrangian constraint algorithm, these additional constraints could not have been due to the generalized Lie symmetry of the action.