NON-NOETHER SYMMETRIES IN HAMILTONIAN DYNAMICAL SYSTEMS

We discuss geometric properties of non-Noether symmetries and their possible applications in integrable Hamiltonian systems. The correspondence between non-Noether symmetries and conservation laws is revisited. It is shown that in regular Hamiltonian systems such symmetries canonically lead to Lax pairs on the algebra of linear operators on the cotangent bundle over the phase space. Relationship between non-Noether symmetries and other widespread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frolicher Nijenhuis geometry is considered. It is proved that the integrals of motion associated with a continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang Baxter type equation. Action of one -parameter group of symmetry on the algebra of integrals of motion is studied and involutivity of group orbits is discussed. Hidden non-Noether symmetries of the Toda chain, Korteweg de Vries equation, Benney system, nonlinear water wave equations and Broer Kaup system are revealed and discussed.