Symmetries And New Conserved Quantities For Relativistic 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 new conserved quantities for the relativistic mechanical systems by the introduction of coordination function. Firstly, we study the new conserved quantities induced by symmetries for relativistic holonomic and nonholonomic mechanical system. Both the conditions of existence of generalized Noether conserved quantity and new generalized Hojman conserved quantity as well as the forms, which are directly led to by Noether symmetry and Lie symmetry and the conditions of existence of generalized Mei conserved quantity as well as the forms, which are indirectly led to by Noether symmetry and Lie symmetry, are given; The conditions of existence and the forms of generalized Noether conserved quantity, new generalized Hojman conserved quantity and generalized Mei conserved quantity directly led to by Mei symmetry are proposed. Secondly, we study the same topic as the first content for relativistic holonomic and nonholonomic mechanical system in phase space. Finally, the researches in this paper are summarized; theoritical researches on symmetries and conserved quantities for relativistic mechanical systems are prospected.