Symmetries of Hamiltonian Equations and ?-Constants of Motion

We consider symmetries and perturbed of canonical Hamiltonian equations motion. Specifically we the case in which exhibit a ?-symmetry under some Lie point vector field. After brief survey relationships between standard existence first integrals, recall definition properties ?-symmetries. show that presence for equations, one can introduce notion ?-constant The leads also to nice useful reduction form equations. then problem is deduced from ?-invariant Lagrangian. illustrate how Lagrangian ?-invariance transferred into context are ?-symmetric. compare partial (Lagrangian) Euler–Lagrange with be obtained equat...