Symmetries of a dynamical system represented by singular Lagrangians

Havelková, Monika

Communications in Mathematics, Tome 20 (2012) no. 1, pp. 23-32.

Résumé

Dynamical properties of singular Lagrangian systems differ from those of classical Lagrangians of the form $L=T-V$. Even less is known about symmetries and conservation laws of such Lagrangians and of their corresponding actions. In this article we study symmetries and conservation laws of a concrete singular Lagrangian system interesting in physics. We solve the problem of determining all point symmetries of the Lagrangian and of its Euler-Lagrange form, i.e. of the action. It is known that every point symmetry of a Lagrangian is a point symmetry of its Euler-Lagrange form, and this of course happens also in our case. We are also interested in the converse statement, namely if to every point symmetry $\xi$ of the Euler-Lagrange form $E$ there exists a Lagrangian $\lambda$ for $E$ such that $\xi$ is a point symmetry of $\lambda$. In the case studied the answer is affirmative, moreover we have found that the corresponding Lagrangians are all of order one.

Détail

Classification : 70H03, 70H33, 70H45
Keywords: singular Lagrangians; Euler-Lagrange form; point symmetry; conservation law; equivalent Lagrangians