Partial Integrability Of Hamiltonian Systems With Homogeneous Potential

In 1834 Hamilton represented the differential equations of classical mechanics, i.e., the Lagrange equationsin the "canonical" formwhere p = (?) is the generalized momentum. H = p(?) - Lis called the "total energy" of the mechanical system. Equation(??) is the canonical form of the famous Hamiltonian systems, H = H(p, q) is the Hamiltonian function, p is the generalized momentum, q is generalized coordinate. The dimensional of q, is the degree of freedom of the Hamiltonian system, without loose of generality, we set it n, then we call system (2) a Hamilton system of n degree freedom.Hamiltonian systems, as a very important field of the nonlinear science, widely exist in the mathematical science, the life sciences as well as social sciences each domain. Especially in the celestial mechanics, and so on granule physics, astronautics science as well as bioengineeringmany models by the Hamiltonian systems (or their perturbation systems) appear, the research of this domain has been therefore prosperous for many years .Early in 19th, the problem of solving Hamiltonian systems(not yet written in the canonicalform) had already been discussed by Bernoulli, Clairaut, D'Alambert and Lagrange in their works connected with application of Newton's idea and principles to various problems of mechanics. At that time, a systems is said to be solvable(integrable), if it can be solved by a finite sequnce of algebraic operations and "quadratures" - indefinite integrals of known functions. One of the famous methods to solve Hamiltonian systems is the Hamilton-Jacobi method developed by Hamilton(1834) and Jacobi(1837), for details see e.g.[9, 28]. Based on this method, Bour and Liouville obtained the famous Liouville theorem in 1855: Theorem 1 (Liouville) Hamiltonian systems with n degree of freedom can be integratedprovided there are n independent integrals in involution.Theorem 1 was developed later by Arnold, and result to the Liouville-Arnold theorem.Theorem 2 (Liouville-Arnold) Assume that the Hamiltonian system (2) defined on the real symplectic manifold M has n functionally independent and pairwise involution first integrals F₁,…,F_n. Let M_a be a non-singular level manifold(i.e., dF₁,…,dF_n∈T^*Mare independent on every point of M_a),M_a = {p∈M;F₁(p) = a₁,…,F+n(p) = a_n},where (a₁,…, a_n) is an n dimensional constant real vector. Then:1. If M_a is compact and connected, then it is a torus M_a (?) Rⁿ/Zⁿ;2. In a neighborhood of the torus M_a there are functions I₁,…,I_n,φ₁,…,φ_n, such thatω= (?)dI_i ? dφ_i and {H,I_j} = 0, (?)j = 1,…,n.According to theorem 2, people usually use the following use following definition for the integrability of Hamiltonian system (2):Definition 1 Hamiltonian system (2) is said to be integrable in Liouville sense (or completely integrable), if it has n functionally independent first integrals in involution.Remark 1 Generally, Whether a Hamiltonian system is completely integrable is not equivalent to that whether ia can be solved by Hamilton-Jacobi method. One of the striking example is the rotation of a rigid body around a fixed point in a gravitational field of distant centers. In this problem Brun [4] found that it is completely integrable while not solved by Hamilton-Jacobi method.On the other hand, many dynamical systems, even if some famous problems, such as N(N≥3) bodies problem, are not solvable( integrable). Early in 1841, Liouville proved the simple linear equation (?) + tx = 0 is unsolvable by quadratures. More exactly, there do not exist a field that contains all solutions of the Liouville equation and that can be obtained from the field of rational functions of t by a sequence of finite algebraic extension, adjoining integrals and exponents of integrals. This caused the mathematicians to start to realize in the universality of the unsoluability of the classical dynamics problems, and thus started about their strict research work on non-integrability. In the present paper, we say the Hamiltonian system (2) is not integrable, if it has no other first integral independent with the Hamiltonianfunction H; say the Hamiltonian system (2) is partial integrable, if is not completely integrable, and not integrable.In 1887, Bruns[5] proved the three bodies problem does not have any other algebraicfirst integrals besides the ten known classical integrals.Poincar6[24] studied the partial integrability of the Hamiltonian system in the neighborhoodof a real periodic curve based on the Monodromy matrix of the variational equation along the real periodic curve:Theorem 3 If there are k first integrals of the Hamiltonian system , independent over the real periodic curve, then k characteristic exponents must be zero. Moreover, If these first integrals are in involuntion, then 2k characteristic exponents must necessarily be zero.Meanwhile, Poincar(?)[23] also studied the problems of integrability of the perturbed systems of completely integrable Hamiltonian systems, obtained the non-integrability of the perturbed Hamiltonian systems under some general hypothesis. There have been many new results about the non-integrability of perturbed Hamiltonian systems, see, e.g, [10, 11, 12].In 1888, 1889å¹´, Kowalevskaya[8] obtained a new case of integrability of the rigid body system with a fixed point, imposing that the the general solution is a meromorphic function of complex time. In fact, , she proved that, except for some particular solutions, the only cases in which the general solution is a meromorphic functionof time are Euler's, Lagrange's and the new case found by her.In 1894, Lyapounov[15] ,by analysis of the variational equation along some known solutions, generalized the Kowalevskaya result and proved that, except some particular solutions,the general solution is single-valued on;y in the above three cases.In 1982, Ziglin [32] proved a non-integrability theorem for complex analytical Hamiltoniansystem. He used the constraints imposed by the existence of some first integrals on the monodromy group of the normal variational equation along some complex integral curve. Ziglin theory has been widely applied in many Hamiltonian systems with lower degree of freedom, e.g.,[33, 27, 31,25].In the 1990s, based on the Ziglin theory and differential Galois theory, used the con- straints imposed by the existence of complete first integrals on the differential Galois group of the variational equation along some complex integral curve, Morales-Ruiz, Ramis [17,18, 19] and so on, established Morales-Ramis theory, for more related introduction and applicationsee, e.g., [16, 20, 21, 22].In this paper, we consider the following hamiltonian system with homogeneous potentialwhere H = (?)p² + V(q), q, p∈Cⁿ is the hamiltonian function , V is a homogeneous function of degree k,k≠0,±2.Lochak's result[14] in 1985 tell us that the system (3)'s K exponents appear in pairs, denoted by (p_i,p_i+n), i = 1,2,…, n. We denote△ρ_i =ρ_i -ρ_i+n. In 1989, Yoshida[29] studied the integrability of system (3) based on the Ziglin Theory, and obtained the result below :Theorem 4 (Yoshida, 1989) if△ρ₁,…,△ρ_n are rational independent, then system (3) does not have any other analytic first integrals independent with H .Let G = {K∈Z?n, K·△ρ= 0}, it is a finite generated subgroup of. Denote the number of generators of G by R(G) Then the following three statements are equivalent:1.△ρ₁,△ρ₂,…,△ρ_n are rational independent;2. R(G) = 0;3.△ρ₁,△ρ₂, ? ? ? ,△ρ_n are integer independent.For simplicity, from now on, we use the terminology of integer independence.The main purpose of present paper is to give a result of partial integrability of system(3).This paper is divided as follows:In Chapter two, we mainly introduce the related definitions and results about the monodromymatrix of linear differential equations, quasi-homogeneous systems, and Zigin theory.In Chaper three, we give the main result and its proof.Note that there is only two cases being able to happen when R(G) = 1: C₁ : There is only one rational number in△ρ₁,…,△ρ_n-1 , and others are integer independentirrational numbers or pure imaginary numbers, for convenience we assume△ρ₁ is rational number;C₂ :△ρ₁,…,△ρ_n-1, are irrational numbers or pure imaginary numbers, and there are two and only two numbers integer dependent, for convenience we assume they are△ρ₁,△ρ₂.The following results are important for the proof of the main theorem:Corollary 1 If the system (3) has an analytic first integral H₁ independent with H, and C₁ is in the case, then the (2.10) has a nontrivial first integralLemma 1 If the system (3) has an analytic first integral H₁ independent with H, anc the second cases (C₂) is in the case, then the (2.10) has a nontrivial first integralLemma 2 (Ziglin lemma) Let M is a meromorphic field on Cⁿ, (?) is a subfield of M. {f₁,…f_n} (?) K are independent, then ,there exist polynomials {P_i(z₁,, z_i)|i = 1,…, n} such that, if g_t = P_i(f₁,…, f_i), then g₁⁰…, g_n⁰ are algebraic independent.Lemma 3 On C(x₁,…, x_n), algebraic independence is equivalent to functional independence.Lemma 4 Let V is a complex 2n dimensional symplectic space. C(V) is the rational function field on V. assume that f₁,…, f_n+1∈C(V)≈C(x₁,…, x_2n) are in involution, then f₁,…,f_n+1 are functional dependent in a open neighborhood U on V.Based on the preliminary work above, it is easy to get the result below :Theorem 5 Assume that the system (3) has two independent analytic first integrals H,H₁. If R(G) = 1, then system (1.3) does not have other analytic first integral which is independent and involutive with H, H₁.