The Theoretical Research About Non-standard Hamiltonian Equation

Analytical mechanics and its research method play a crucial role in physics,mechanics and engineering.For analytical dynamics,its greatest benefit is to propose a set of methods which establish a series of mechanical systems based on the Hamiltonian principle.The analytical mechanics method not only applies to classical mechanics problems.At the same time,it plays an important role in the development of electrodynamics,statistical mechanics,quantum mechanics,nonlinear science and some modern physics theories.This paper is based on the Hamilton principle.It introduces an exponential form of Hamiltonian function and constructs the non-standard Hamiltonian functional,and then the nonstandard Hamiltonian equation is obtained by using the variation principle.This equation is different from the traditional Hamiltonian equation.Its Hamiltonian function is not the sum of kinetic energy and potential energy and its equation is more complex than the traditional Hamiltonian equation.But under some special conditions,it can be reduced to the standard Hamiltonian equation.So the standard Hamiltonian equation is a special case of the non-standard Hamiltonian equation.At the same time,the non-standard Hamiltonian equation is also completely different from the standard Hamiltonian equation.For non-standard Hamiltonian equation,even if the Hamiltonian function does not explicitly contain time,it is not necessarily conserved.If the non-standard Hamiltonian function does not explicitly contain a generalized coordinate,the generalized momentum corresponding to its generalized coordinates is not necessarily conserved.Only under special conditions,the generalized momentum corresponding to the Hamiltonian or the generalized coordinates that do not contain the time is conserved.Non-standard Hamiltonian functions exist in some complex dynamics systems such as physics,mechanics and engineering.The non-standard dynamics system is used to study some special complex nonlinear dynamic system problems,and the results are simple and easy to operate.Non-standard Hamiltonian system provides us with a new class of dynamical system and can be applied to nonlinear dynamic system,the dissipative dynamic system,the classical theory of quantization problem and cosmology,etc.Firstly,the development and research status of variation method and non-standard dynamics systems are introduced.Secondly,I introduce the classical variation principle,the Lagrange equation and Hamiltonian equation of the standard form,and the Lagrange equation theory and its application of non-standard form.In the end,the dynamic properties of the non-standard Hamiltonian equation are discussed in detail,and its application in nonlinear mechanics is illustrated.It can be seen from the examples that the non-standard Hamiltonian equation plays an important role in solving nonlinear dynamic problems.For some of the more complex dynamic problems,the introduction of non-standard Hamiltonian equations and non-standard Lagrange equations can sometimes make the problem simple and reduce the computational intensity.Therefore,the non-standard Hamiltonian system is an important dynamic system,which is worthy of further study.