The Lie Symmetry Of Constrainted Hamilton Systems And Its Application In Field Theory

In general,there are two kinds of constrained in mechanical systems.One is a regular system with constraints imposed by the outside,and the other is the constraint Hamilton system.The former is a system described by a formal Lagrangian,which is subject to additional binding,the latter being a singular system described by a singular Lagrangian whose constraints refer to the relationship between regular variables in a phase space.In addition,we know that a dynamical system can be described by both Lagrange and Hamilton forms.For a normal system,the regular variables are independent of each other,when the Lagrange description in configuration space transitions to the Hamiltonian description in phase space.However for singular systems,there is a relationship between regular variables,also known as the system’s inherent constraints,we call the constrained Hamilton system.In addition,quantum field theory is a hot research field today,and most of the systems in quantum field theory are singular.Therefore,this paper studies the symmetry and its application of constrained Hamilton system.In this paper,the Lie symmetries and conserved quantities of constrained Hamiltonian systems are proposed.The Noether conserved and non-Noether conserved quantities,(Mei conserved quantities and Hojman conserved quantities)are given.In addition,the symmetry of the constrained Hamilton system based on the integral factor method is studied.Finally,the two methods are extended to the field theory.The application of the integral factor method and the Lie symmetry method in field theory are studied before and after.The symmetries of gauge invariant of self coupled field and scalar field coupling Chern-Simons,and seeks its conserved quantity.The research contents of this paper include the following aspects.First,the regular equations of motion and constrained equations of constrained Hamiltonian systems are given.According to the invariant of the motion differential equations and the inherent constraints for the constraint Hamilton systems under infinitesimal transformations,the deterministic equations,the constrained equationsand the additional constrained equations are established.Then the system structural equation is given to obtain the conserved quantity of the system,and the Noether conserved quantity is given according to the Noether identity.Finally,we further study whether the infinitesimal generators satisfy the constrained equation and the additional constrained equation,then the common Lie symmetry,weak Lie symmetries and strong Lie symmetries are obtain.Second,we give the relation between the Lie symmetry and the non-Noether symmetry of the constrained Hamilton system,and derive the non-Noether conserved quantity of the system from two aspects(Mei conserved quantity and Hojman conserved quantity).On the one hand,according to the invariant of the dynamics function for the system under the infinitesimal transformation,then the Mei symmetry and conserved quantities of constrained Hamilton systems is presented,on the other hand,On the other hand,starting from the differential equations of the system,the special Lie determination equations of the constrained Hamilton system are deduced from the special Lie symmetries of the time invariable.the conserved quantities have proved to be non Noether conserved quantity.Thirdly,the integral factor and symmetry of constrained Hamiltonian system is studied.The canonical motion equations of constrained Hamiltonian system are given.The integral factor and conservation theorem of constrained Hamiltonian system are constructed.The generalized Killing equation of the system is established.Finally,we can find the integral factor and the unknown function from generalized killing equation.Finally,the conservation of the system is given by the conservation theorem.Fourthly,we establish the canonical Hamiltonian equation of the field theory in phase space,then the integral factor and conservation theorem of the field theory system are given,furthermore,construct the generalized killing equation of the field theory system,and finally give the conserved quantity of the field theory system.Taking the gauge invariant of self coupled field as an example,the feasibility and advantages of the integral factor method are illustrated.Fifthly,according to the symmetry theory of constrained Hamilton system,theLie symmetries and the conserved quantities of the field theory are studied.The canonical equations and constrained equations of the field theory system are given.According to the invariant of the canonical equations and constraint equation,the determining equation,restriction equation and additional restriction equation are constructed,then the structural equations and conservation theorems of the field theory system are constructed.The infinitesimal generators deduced from the deterministic equations,furthermore,the general Lie symmetry,weak Lie symmetry and strong Lie symmetry are discussed,and the general conserved quantity,weak Lie conserved quantity and strong Lie conserved quantity of field theory system are obtained.