Isotropic submanifolds and the inverse problem for mechanical constrained systems

摘要

The inverse problem of the calculus of variations consists in determining ifthe solutions of a given system of second order differential equationscorrespond with the solutions of the Euler-Lagrange equations for some regularLagrangian. This problem in the general version remains unsolved. Here, wecontribute to it with a novel description in terms of Lagrangian submanifoldsof a symplectic manifold, also valid under some adaptation for thenon-autonomous version. One of the advantages of this new point of view is thatwe can easily extend our description to the study of the inverse problem of thecalculus of variations for second order systems along submanifolds. In thiscase, instead of Lagrangian submanifolds we will use isotropic submanifolds,covering both the nonholonomic and holonomic constraints for autonomous andnon-autonomous systems as particular examples. Moreover, we use symplectictechniques to extend these isotropic submanifolds to Lagrangian ones, allowingus to describe the constrained solutions as solutions of a variational problemnow without constraints. Mechanical examples such as the rolling disk areprovided to illustrate the main results.