Symmetries of a dynamical system represented by singular Lagrangians. (English) Zbl 1271.70029

Summary: 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.


70H03 Lagrange’s equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70H45 Constrained dynamics, Dirac’s theory of constraints
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