Nonlocal constants of motion in Lagrangian Dynamics of any order

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.