Momentum-Dependent Symmetries and Non-Noether Conserved Quantities for Nonconservative Hamilton Systems

In this letter, based on the infinitesimal transformations with respect to the generalized coordinates and generalized momentums, we obtain the definition, determining equations and structure equation of the momentum-dependent symmetry for the systems. This study directly leads to the non- Noether type conserved quantity for the systems. Further we also give the inverse issue of the momentum-dependent symmetries of the systems. However, a theory of momentum-dependent symmetries of the nonconservative Hamiltonian systems is established. Finally, an example is discussed to illustrate the results.