The Renormalization Group Equations For A Class Of Hamiltonian Systems

Ordinary differential equation is an important branch of mathematics,not only in physics,engineering,biology,meteorology and other disciplines have important applications,but also in geometry,function theory,algebra,variational method,topology,functional analysis,harmonic analysis and other important branches of mathematics has played an important role in the development.The establishment,perfection and development of the theory of ordinary differential equations,not only to promote the relevant branches of mathematics to flourish,but also to expand the scope of the study and methods of these mathematical branch.As early as the 1870 s,it is found that only a very small number of equations can be expressed by elementary functions,this makes people not fully focused on the concrete solution of the equation,but rather directly according to the structure of the equation to explore some of the nature of the solution to the problem,for example: the existence of the solution,uniqueness,stability and so on,and directly lead to the differential equation qualitative theory and geometric theory and the establishment,development and perfection of the power system.While the classic solution to the problem,people are more concerned about the study of practical problems,rather than focusing solely on the exact solution of a system or equation.For example,a mathematical theory model that comes down to practical problems in life,it is often a nonlinear or higher order differential equation,and on this basis with a non-linear boundary conditions or initial conditions,and some of the problems of the border shape is very complex,uncertain,for most of these problems is not accurate solution,on the other hand,even if some of the equation can get its exact solution,but the exact form of expression is too complex to apply to practical problems,so people turn to seek the approximate solution or numerical solution of the problem,or the combination of the two.In order to approximate the differential equation,mathematicians,physicists after many years of work together,in the course of research on various specific perturbation problems,the establishment and development of a number of important skills and methods,such as averaging method,multi-scale method,matching asymptotic expansion method,telescopic coordinate method,WKB method,central manifold method and so on,these methods are collectively referred to as perturbation methods,the basic idea is to express the exact solution of the problem with a certain number of perturbation expansion.Since the middle of the last century,the development of perturbation method is very fast,it is widely used in physics,chemistry,engineering,astronomy and many other nature disciplines in various fields,and get some important results.But people also found that these methods with some limitations,for example: multi-scale method is not easy to determine the time scale,although the asymptotic expansion method is applicable to both linear and non-linear problems,but the location and thickness of the boundary layer is generally not easy to determine,at the same time,there may be a fractional power of small parameters,in addition,but also through the middle of the expansion of the internal expansion and external expansion to match and so on.It is particularly important to find a more convenient and effective way to overcome the limitations of traditional deterrence method.In the 1940 s,Bethe and other people proposed the classical theory of renormalization in the study of photon propagation.Wil-son applied the renormalization group method to study the problems in quantum field theory and statistical physics,and made important research results,based on these results he won the 1982 Nobel Prize in Physics,this makes the renormalization group method more concerned by mathematicians and physicists.Then Bricmont and other people proposed techniques such as charge recombination,mass renormalization,coupling constant renormalization,and wave function re-normalization,cleverly overcome the limitations of the traditional perturbation methods,the solution of the complex singular perturbation problem is transformed into solve the problem of simple equations-the renormalization equation.In recent years,Chen,Golden-feld and Oono developed and refined the renormalization group method,on the one hand,they studied several important singular perturbation problems,we get the unanimous and asymptotic expansion of these problems,on the other hand,it explores the relationship between the renormalization group method and the classical singular perturbation technique.As we all know,hamiltonian system is a very important ordinary differential equation,such systems in many research areas,especially in celestial mechanics and physics have important applications,naturally,the renormalization group method is also applied to the problem of Hamiltonian system.In 1998,Yamaguchi and Nambu studied a class of two degrees of freedom Hamiltonian systems using the renormalization group method.They proved: if the second-order positive group equation of the Hamiltonian system is a Hamiltonian system,then the original Hamiltonian system and the renormalization group equations are integrable.Specifically,they consider the following Hamiltonian systemε is a small parameter,the potential function V（q₁,q₂）is q₁ and q₂quadratic or cubic homogeneous polynomials,get the following results:Theorem 0.1 The renormalization group equation（RGE）of the systerm（1）is a Hamiltonian system up to the second order of ε,then the original system（1）and its renormalization group equations are integrable.Subsequently,they further consider the potential energy V（q₁,q₂）is the analytic function,and only the case of q₁ and q₂even term,get the following results:Theorem 0.2 The renormalization group equation（RGE）of the systerm（1）is a Hamiltonian system up to the second order of ε if and only if the original system（1）is separable by Cartesian coordinates,that is,rotation of the coordinates q₁ and q₂.In the text,we will use the renormalization group method to study a class of two degrees of freedom Hamiltonian system.consider the following Hamiltonian systemThe main results of this paper are as follows:Theorem 0.3 The O（ε）renormalization group equation of the Hamiltonian system（2）is The structure of this paper is as follows:In the second chapter,we take a single degree of freedom Hamiltonian system as an example,brief description of the renormalization group approach.In the third chapter,We first derive the renormalization group equation of a class of two degrees of freedom Hamiltonian system,then the equation is a Hamiltonian system.