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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.

MSC:

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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