Symmetry Reduction Of Constrainedmechanical Systems

Symmetry reduction theory is an important theory of analytical mechanics in recent years.Symmetry reduction is the sublimation of symmetry theory.Symmetry reduction theory is to reduce the system by using some symmetry and its conserved quantity of the dynamical system.Therefore,the study of the dynamic of the original system is limited to the dynamical problems in the low-dimensional space,at the same time the nonlinear dynamical problem in the original space is also transformed into the linear dynamical problem on the dual space of Lie algebra.In this paper,the symmetry reduction of the constrained mechanical system is studied,including the following research contents:Firstly,the Lagrange equation and the Hamiltonian equation derived from the ideal constraint assumption and the D' Alembert principle in analytical mechanics are briefly introduced.Secondly,the symplectic manifold and its various submanifolds are introduced.The theory of symplectic reduction and the application of the theory to Lie Group and Hamiltonian system is studied in detail.An example is given to the application of the symmetry theory to the constrained Hamiltonian system,and compared with the Hamiltonian equation derived from the ideal constraint assumption and D' Alembert's principle in analytical mechanics.It is found that the Frobenius integrability theorem used in deriving the Hamiltonian equation by the theory of symplectic reduction is equivalent to the ideal constraint assumption.Finally,the symmetry reduction problem of the constrained system is studied.By means of the momentum mapping,the theory of the symplectic reduction and its dynamics are given.The symmetry reduction problem of the complete constrained system and the nonholonomic constrained mechanical system is studied in detail.The main research includes: studying the Routh reduction problem of the complete system and the symmetry reduction of the classical two-body problem in the framework of the symmetry the theory of the symplectic reduction;And the theory of symplecticreduction is applied to the study of symmetry reduction of Chaplygin nonholonomic systems.