Symmetries of second order potential differential systems

We characterize the family of second order potential differential systems, with n degrees of freedom, via their symmetries. Firstly, we calculate explicitly the equivalence Lie algebra and the weak equivalence Lie algebra. It is shown that the equivalence Lie algebra has the dimension n + 4 + n(n− 1) 2 whereas the weak equivalence Lie algebra is infinitedimensional. The later contains strictly the former. Secondly, we investigate the Lie-point symmetry structure. We start with a quadratic potential, and we provide an analysis relying on the spectral theorem. In the case of non-quadratic potentials, we establish the conditions for the existence of additional symmetries, deriving the classifying conditions. These conditions are greatly simplified under the action of the equivalence group. Finally we show how the Lie point symmetries can be obtained using (weak) equivalence transformations and we give an example where the existence of symmetries can be used to prove integrability. Mathematics Subject Classification: 76M60, 70G65, 37J15, 58D19, 58J70.