arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2301.08396math-phgr-qchep-thmath.MPphysics.class-ph
收藏

广义李对称与几乎正则拉格朗日量：对称性与动力学之间的联系

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

Achilles D. Speliotopoulos

备注：
34 pages with one table. Published paper is open access, and can be found either at the Journal of Physics Communications website or at the DOI below. This is a follow-up paper to "Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics", DOI: 10.1088/2399-6528/ab923c. arXiv admin note: text overlap with arXiv:2006.02614
发表信息：
J. Phys. Commun. 7 (2023) 015001
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

研究了几乎正则拉格朗日量的广义李对称性，并确定了它们对动力系统演化的影响。它被发现，如果行动有一个广义李对称，然后拉格朗日必然是奇异的;相反的是不正确的，因为我们显示了一个具体的例子。还发现作用量的广义Lie对称性是Euler-Lagrange运动方程的广义Lie对称性的一个李群。反过来也是不正确的，有一些系统的欧拉-拉格朗日运动方程具有广义的李对称性，而作用量却不具有，我们再一次通过一个具体的例子说明了这一点。最重要的是，它表明，每个广义李对称的行动贡献一个任意函数的动力系统的演变。这种对称的数量给出了拉格朗日相空间中任何一组允许的初始数据所产生的曲线族的维数的下限。此外，如果在应用拉格朗日约束算法的过程中引入二阶或高阶拉格朗日约束，则这些额外的约束不可能是由于作用的广义李对称性。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑