Jacobi multipliers, non-local symmetries and nonlinear oscillators

摘要

Constants of motion, Lagrangians and Hamiltonians admitted by a family ofrelevant nonlinear oscillators are derived using a geometric formalism. Thetheory of the Jacobi last multiplier allows us to find Lagrangian descriptionsand constants of the motion. An application of the jet bundle formulation ofsymmetries of differential equations is presented in the second part of thepaper. After a short review of the general formalism, the particular case ofnon-local symmetries is studied in detail by making use of an extendedformalism. The theory is related to some results previously obtained byKrasil'shchi, Vinogradov and coworkers. Finally the existence of non-localsymmetries for such two nonlinear oscillators is proved.