Symmetries And Conserved Quantities For Nambu Mechanical Systems

The researches on symmetries and conserved quantities have important theoretical value and practical significance. They have been a hot research field for mathematics, physics and mechanics. There are three modern symmetrical methods for analytical mechanics: Noether symmetry, Lie symmetry and Mei symmetry. The three kinds of symmetries can lead to Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity. In this paper, we study the symmetries and related conserved quantities for the Nambu mechanical systems. Firstly, we study the new conserved quantities induced by symmetries for classical Nambu mechanical system. Both the conditions for the existence of a new Hojman conserved quantity as well as the forms, which are led to by Lie symmetry directly and Mei symmetry undirectly, are given. The conditions that the system have both Lie and Mei symmetries are proposed. Secondly, we expand the study field of classical Nambu system to generalized Nambu system by introducing the conception of quasi-Nambu-Hamiltonian and insidual term. Then we study the Lie and Mei symmetris in this generalized Nambu system and obtain a new Hojman conserved quantity as well as the Hojman and Kai conserved quantities. Thirdly, we study the situation that the generalized Nambu system is under some constraints. The Lie and Mei symmetry and the conserved quantities, including a new Hojman conserved quantity, induced by the symmetries, directly or undirectly, are studied Forthly, the symmetris and the conserved quantities after the three systems perturbed are studied. And the related adiabatic invariants for the systems are obtained, including Hojman type, Kai type and new Hojman adiabatic invariant, which are all z order. Finially, the researches in this paper are summarized and theoritical researches on symmetries and conserved quantities for Nambu mechanical systems are prospected.