Separability of completely integrable dynamical systems admitting alternative Lagrangian descriptions (Q1063300)

It is shown that whenever a (1,1)-type tensor field, defined through two alternative Lagrangians, implies complete integrability for a second- order Lagrangian vector field \(\Gamma\) on the tangent bundle of a given configuration manifold, then a bundle atlas can be found in which both \(\Gamma\) and a class of equivalent Lagrangians are completely separated into a sum of second-order vector fields and Lagrangians, each one corresponding to a single degree of freedom.