  We describe a recipe to generate "nonlocal" constants of motion for ODE
Lagrangian systems. As a sample application, we recall a nonlocal constant of
motion for dissipative mechanical systems, from which we can deduce global
existence and estimates of solutions under fairly general assumptions. Then we
review a generalization to Euler-Lagrange ODEs of order higher than two,
leading to first integrals for the Pais-Uhlenbeck oscillator and other systems.
Future developments may include adaptations of the theory to Euler-Lagrange
PDEs.
