G. Chavchanidze

Non-Noether Symmetries in Hamiltonian Dynamical Systems

abstract:
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 Frölicher-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.

Mathematics Subject Classification: 70H33, 70H06, 58J70, 53Z05, 35A30

Key words and phrases: Non-Noether symmetry, Conservation law, bi-Hamiltonian system, Bidifferential calculus, Lax pair, Frölicher-Nijenhuis operator, Korteweg-de Vries equation, Broer-Kaup system, Benney system, Toda chain