The Theory And Application Research On Power Law Hamiltonian Equation

There are many complicated nonlinear dynamic problems in nature,and for describing nonlinear dynamic systems and dissipative dynamic systems.The power function form of the Hamiltonian system has significant advantages compared to its traditional form of system.When solving relatively complex nonlinear problems,applying the power function form of the Hamiltonian equation can simplify the problem and make it easy to calculate.Therefore,the power function form of the Hamiltonian equation has important research significance.In this paper,the Hamiltonian equation of power function form is studied,namely the power law Hamiltonian equation.Its form,nature and application are explored.And also studies the symmetry and conserved quantity of the power law Hamiltonian dynamic system.In this paper,firstly,the power law Hamiltonian action is defined.Then the power law Hamiltonian equation is obtained by using to the Hamiltonian principle and the variation method.This equation has the biggest characteristic,adjustable parameters ?.Different motion trajectories of objects can be obtained by adjusting the?.The power law Hamiltonian equation has a complicated equation,however,when it is under a certain constraint.The power law Hamiltonian equation can be reduced to the standard form,which can also be called the traditional form or the natural form.Different forms of equation are obtained under different constraints.This paper also obtains some unique properties of the power law Hamiltonian equation under different constraints.This function is not always conserved when its corresponding function does not contain time.If this function is to be kept conserved,certain constraints are required.This same applies to generalized momentum.If the corresponding generalized coordinates do not appear in this function,then the momentum can’t be conserved except under certain conditions.In this paper,a part of the theoretical application of the power law Hamiltonian equation is also introduced.In these application examples,it is also verified that the properties of the function corresponding to this equation obtained in this paper are correct.In this paper,the Noether symmetry and conserved quantity of the power law Hamiltonian dynamic system are used as the theoretical basis.Firstly,the Killing equation of this system is established,and the Killing equation is used to solve some problems that can’t be solved by Noether symmetry criterion and Noether theorem.Secondly,based on the previous research,the Lie symmetry of the dynamic system based on the power law Hamiltonian function is given,and the proof is given.Finally,it is verified by an example.The power-law Hamiltonian equation obtained in this paper has very good theory and application.This equation can simplify the process when dealing with problems,and the obtained results are also easy to understand,so this equation has broad application prospects,and it is worth exploring the application in more depth and it is worth extending.