Symplectic Geometry Algorithm For Lotka-Volterra System

Hamiltonian mechanics,Newtonian mechanics and Lagrangian mechanics are three forms of classical mechanics.These different mathematical forms express the same phys-ical laws.All real,negligible dissipation physical processes can be expressed in the form of Hamiltonian systems.Hamiltonian systems in plasma,Astromechanics and partial differential equations are widely used in many fields.At present,Hamiltonian systems has become an indispensable mathematical tool in the study of physical theory,and the symplectic geometry algorithm is the basis of Hamiltonian systems.Hamiltonian systems is an important system of a dynamic system with a unique symplectic structure,In the field of computational mathematics,symplectic algorithm is a relatively active branch.and it is a computing method that can maintain stability for a long time.Of course,one of the most important means of constructing the symplectic difference scheme of the sym-plectic algorithm is to use the variational method.The variational method is put forward by physicists,and the conclusion is given by mathematicians.Therefore,the variational method is widely used in various fieldsFirstly,this paper introduces Hamiltonian systems and symplectic geometry algo-rithm,the research background and the current situation of the variational method and Lotka Volterra system.Secondly,it introduces the classical one-step method of numerical solution of ordinary differential equations,Hamiltonian systems,and symplectic geom-etry algorithm,the theoretical basis of the variational method and the Lotka Volterra model.Finally,by studying the Lotka Volterra system,it shows that:compared with the non-symplectic algorithm,the Symplectic algorithm has obvious advantages in efficiency,stability,long-time tracking ability and long-term approximate invariantsFor the Lotka-Volterra system in the following form(?) The classical symplectic algorithm is derived by using the idea of the variational method.Compared to the phase orbit and energy error obtained by the symplectic algorithm and non-symplectic algorithm,respectively.Experiment result illustrates that the symplectic algorithm not only can keep the phase orbit well in the long time calculation,but also can make the energy error conserved approximately.