Non-Noether symmetries in Hamiltonian Dynamical Systems

Details Non-Noether symmetries in Hamiltonian Dynamical Systems Non-Noether symmetries in Hamiltonian Dynamical Systems

2004-05-03

arxiv.org/abs/math-ph/0405003v1

Physics

Mathematical Physics

LaTeX 2e, 75 pages, no figures

Scientific paper

We discuss geometric properties of non-Noether symmetries and their possible applications in integrable Hamiltonian systems. Correspondence between non-Noether symmetries and conservation laws is revisited. It is shown that in regular Hamiltonian systems such a symmetries canonically lead to a Lax pairs on the algebra of linear operators on cotangent bundle over the phase space. Relationship between the non-Noether symmetries and other wide spread 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 the 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 algebra of integrals of motion is studied and involutivity of group orbits is discussed. Hidden non-Noether symmetries of Toda chain, nonlinear Schrodinger equation, Korteweg-de Vries equations, Benney system, nonlinear water wave equations and Broer-Kaup system are revealed and discussed.

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