On the inverse problem with symmetries, and the appearance of cohomologies in classical Lagrangian dynamics
We review here some aspects of Inverse Problem of the Calculus of Variations when point symmetries of the dynamics are taken into account and one tries to implement them at the level of the Lagrangian description of a dynamical system. Conditions under which one can define strictly invariant Lagrangians are reviewed and shown to be linked to the triviality of a two-cocycle on the Lie algebra of the relevant symmetry group. When the two-cocycle is not trivial, we discuss how one can recover invariant Lagrangian descriptions by enlarging the carrier space and by employing, on the enlarged space, central extensions of the symmetry group (or of its Lie algebra). We also introduce a Lagrangian momentum map, and show that its equivariance is again connected with the (possible) invariance of the Lagrangian. We finally point out the close analogy between the mathematical structures arising in this analysis of classical dynamical systems and similar structures arising in the study of anomalies in quantum field theory (Wess-Zumino terms).