Integrability Of Analytic Hamiltionian System Implies Convergent Birkhoff Normalization

In nature, many mathematical model can be expressed by Hamiltonian systems. therefore, it is an important thing to explore the dynamical activity of Hamilton sys-tems. For completely integrable Hamilton system, we can solve it to get its indepen-dently first integrals. Then we can understand the activity of the whole system, so the integrability of analytical Hamilton system is a property which is worthwhile to be-ing explored deeply. Because Birkhoff normal form can be used to simplify dynamical system, and it is applied successfully in many aspects, such as bifurcation of periodic orbits, KAM theory, stability problem and so on. Therefore, Birkhoff normal is a very important tool to research dynamical systems.In a locally symplectic coordinate system, if the Hamiltonian and the semisimple part of its quadratic part are Poisson commuting, then we call the Hamiltonian a Birkhoff normal form, the transformation a Birkhoff normalization. In general, we consider the convergence of the transformation.(It is a power series formally.) The analytically integrable Hamiltonian is a complete set of locally analytic, functionally independent, first integrals in involution.The problems have been studied by many mathematicians, including Poincare, Birkhoff, Siegel, Morser, Bruno and so on. Now that Birkhoff normal form and first integrals can be used to simplify and solve Hamilton system separately, these two must be closely related.In this paper, we consider the Hamiltonian system with n degrees of freedom which is analytic in a neighborhood of the origin, In fact, for nonresonance case, in 1927 Birkhoff [17] proved that convergent Birkhoff normalization can imply integrability. The inverse has been a difficult problem, in 1964 Russmann [27] proved this problem under an additional nondegeneracy condition to the momentum map with two degrees of freedom; then in 1978 Vey [28] gave a proof for any number of degrees of freedom; finally, in 1989 H.Iton [22] solved the problem without any additional condition.For simple resonance case, it is obvious that convergent Birkoff imply analytic integrability. The inverse was proved by H.Ito [10] in 1992 and Kappeler, Kodama, Nemethi [24] in 1999.when the degrees of resonance are no less than two, convergent Birkhoff does not imply integrability. The reason is that in this case the Birkhoff normal form will generate n - q+1 first integrals in involution, where n is the degrees of freedom, q is the degrees of resonance. But other first integrals do not exist, not even formal ones. The inverse was proved in 2005 by Nguyen Tien Zung [29] who used a new geometric approach without any restriction of degrees of freedom.This paper is an overview that analytic integrability imply convergent Birkhoff in Hamilton system.Firstly, when q= 0, there is an symplectic transformationtake the Hamiltonian H to the formPeople concerned that when(?) is convergent, in H.Ito [22] gave the main theorem:THEOREM 1. Let the origin be a nonresonance equilibrium point of the system (1) and assume that the Hamiltonian H(z)(z= (x, y)) is analytic in a neighbourhood of the origin. Assume that in addition to G1= H the system (1) possesses n - 1 analytic integrals G2(z),..., Gn(z) near the origin such that G1, G2,..., Gn are functionally independent. Then there exists an analytic canonical transformation z=(?)(ζ) (ζ= (ζ,η)) near the origin such that (?)(0)= 0 and Gk(?)(k= 1,..., n) are analytic functions of n variablesζlηl(l= 1,..., n).We omitted the proof of this theorem, the main idea is rapidly convergent iteration method. Secondly, when q= 1, H has the power seriesH= H2+H3+..., (2)where the quadratic part H2 has the decompositionH2= S+N,{S,N}= 0, (3)where S and N is the semisimple part and the nilpotent part of H2 respectively.This case the main theorem is:THEOREM 2. Let H be a holomorphic function of z= (x, y)∈Cn×Cn in the form (2) and let the origin z= 0 be an equilibrium point of simple resonance for the vector field XH. Assume that there exist n holomorphic integrals of HH in a neighbourhood of the origin which are Poisson commuting and functionally independent. Then there exists a holomorphic symplectic transformation z=(?)(ζ) such that H o(?)(ζ) is in S-normal form, where S is e semisimple part of H2 given by (3) with z replaced byζ= (ζ,η)∈Cn×CnBesides, in order to solve the case q≥2, a new geometric approach had been introduced by Nguyen Tien Zung [29]. It was proved that the analytic integrability imply convergent Birkhoff. First, we show the existence of 1-cycles lying on a level set of the momentum map. Then we search the required action.Finally, we pointed that the nonlinear wave equation is an infinite Hamilton sys-tem. We set the Bikhoff normal form theorem for nonlinear PDEs, and applied this theorem to the wave equation.