Canonicalization And Symmetry Theories Of The Constrained Hamiltonian System

In this paper,the canonicalization of the constrained Hamiltonian system and its symmetry theories are studied.Under the Legendre transformation,the singular Lagrangian system can be transformed into phase space and described by Hamiltonian canonical variables,there are inherent constraints between canonical variables,namely constrained Hamiltonian system.When the system has constraints,the motion equations are non-canonical,thus many useful properties and algorithms can not be applied directly.In order to make the motion equations of the constrained Hamiltonian system have canonical form.In this paper,we define a variable transformation,new canonical variables for constrained Hamiltonian system can be given through a series of derivations,then the motion equations have canonical form under the new variables.It is more convenient to study the constrained Hamiltonian system.After the canonicalization is realized,the symmetry theories of the general constrained mechanical system can be applied directly to the constrained Hamiltonian system.In this paper,the Noether symmetry and Lie symmetry of the constrained Hamiltonian system under the canonicalization method are given,it is easy to get the conserved quantities of the system.Then the canonicalization method and the symmetry theories of the constrained Hamiltonian system are applied to the field theory system,and the good results are obtained.At the end of each chapter,examples are given to illustrate the applications of the results.The main research contents and achievements are listed as follows.Firstly,the fundamental theory of the constrained Hamiltonian system is introduced.It shows how the constraints appear when the singular Lagrangian system is converted to phase space.The meaning of the primary constraint,the secondary constraint,the first class constraint and the second class constraint are clarified,which provide the theoretical preparation for the canonicalization method below.Secondly,the canonicalization method of the constrained Hamiltonian system is discussed.The concrete idea is to construct a variable transformation,new canonical variables for the constrained Hamiltonian system can be given through a series of derivations,then the motion equations have canonical form under the new variables.After that,two examples are given to verify the feasibility of this method.On the basis of canonicalization,the Noether symmetry theory of the constrained Hamiltonian system is discussed.After the canonicalization is realized,the Noether symmetry theory of the general Hamiltonian system can be applied directly to the constrained Hamiltonian system.This provides a new method for studying the Noether symmetry of the constrained Hamiltonian system.The contents of this discussion are the variation in the constrained Hamiltonian system,the Noether symmetrical transformation,Noether quasi-symmetrical transformation,generalized Noether quasi-symmetrical transformation,killing equation and Noether theorem of the constrained Hamiltonian system.Then the Lie symmetry of the constrained Hamiltonian system is studied.After the canonicalization is realized,the Lie symmetry theory of the general constrained mechanical system can be applied directly to the constrained Hamiltonian system.The specific contents include the establishment of the motion equations of the constrained Hamiltonian system,the Lie symmetrical transformations,the determining equations,the restriction equations of Lie symmetry,the structural equations of Lie symmetry and the existent conservation laws of the constrained Hamiltonian system through the infinitesimal transformations are studied.In addition,the relationship between the Noether symmetry and the Lie symmetry of the constrained Hamiltonian system is also discussed.In the end,two examples are given to study the application of the symmetry theories of the constrained Hamiltonian system.Finally,the canonicalization method and the symmetry theories of the constrained Hamiltonian system are applied to the field theory system.The Hamilton equations of the field theory system are given,the canonicalization of the Hamilton equations in the field theory system can be realized by the former canonicalization method.Then the symmetry theories and corresponding conserved quantities of the field theory system are given.At the end,the application of theories are verified by the example of the motion of charged particles in the electromagnetic field.