Morales-Ramis Theory And Its Applications

As is well known that, the problem of solving differential equations is involved in the change rules of various aspects such as greatly to the astronomy photostar movement, slightly to the physical microscopic quantum movement, in as well as real life's engineering mechanics, the biological population evolution, the chemical reaction, the social economy fluctuation situation, the signaling and so on. Therefore, from the date of birth of the differential equations, this problem has been valued by each time mathematicians and physicists, and becomes one of the glorious and important problems in research of differential equations.During the initial period of the development of differential equations, people had been trying to seeking the explicit solutions . However, in 1841, the famous mathematician Liouville discovered that many simple differential equations, such as Ricati equation can not be solved by method of elementary integrals[33]. This has caused about the research of the integrability of differential equations. But until now, there is not an explicit identical definition about the integrability of the differential equation.In this paper, we only consider the analytic dynamical systems defined on the complex field. Then, there must be at least two class of systems, i.e., the linear systems and Hamiltonian systems, about which, the definition of the integrability is explicit.Consider linear differential systemwherey=(y₁,…,y_m)^T,K is a differential field.Mat_m(K) is the set of all the m×m matrices on K.Definition 1 The linear differential system (1) is said to be integrable, or solvable, in sense of Liouville, if its solutions can be obtained by integrals, exponent integrals, algebraic elements and the finite composition of them based on the coefficient field.For the analytic autonomous system where v(z) is m dimensional vector-valued analytic function, we have the following definition:Definition 2 Let U(?)C^m is an open set. A functionΦ:U→C is called a first integral of system (2), ifΦ(z) along every solution curve of (2) are all constants. IfΦ(z) is differentiable, this condition is equivalent toEspecially,ifΦ(z) is analytic (or meromorphic) function with respect to z and satisfies (3), thenΦ(z) is called the analytic (or meromorphic) first integral of system (2).In this paper, we mainly consider the n degree of freedom Hamiltonian system on the 2n dimensional complex symplectic manifold Mwhere(q, p) =(q₁,…,q_n,p₁,…,p_n)∈M.H=H(q,p) is the corresponding Hamiltonian function.Definition 3 The Hamiltonian system (4) on the symplectic manifold M is integrable in sense of Liouville, or completely integrable, if it has n functional independent first integrals f₁=H,f₂,…,f_n in involution on M.If system (4) does not have other first integrals functional independent with H, then we say id is non-integrable.In the mid-1990s Morales-Ruiz, Ramis[12, 13, 14]及Baider, Churchill, Rod, Singer[2] developed a famous integrability theory, called Morales-Ramis theory, which is very effective on the research of integrability and non-integrability of Hamiltonian systems. There have been many applications on Morales-Ramis theory, see [11,15, 16, 17]and so on.The present paper will briefly introduce the Morales-Ramis theory and its further develops, and apply these develops to some famous examples, obtain some new non-integrability results.Assume the Hamiltonian system (4) has a nontrivial solution Z(t), it determines a sub-Riemann surfaceΓ[41].then, the variational equation of system (4) alongΓisThen we haveTheorem 1 (Morales-Ramis). Assume that Hamiltonian system(4) has n functional independent first integrals in involution on the neighborhood of the Riemann surfaceΓdetermined by the particular nontrivial solution. Then the identity component of the differential Galois group of the variational equation of system (4) alongΓis Abelian, i.e., commute.Especially, we consider the Hamiltonian system with homogeneous potentialIt is easy to see that, assume equation c=(?)(c) has a nontrivial solution on Cⁿ,then system (6) has a nontrivial solution Z(t)=(φ(t)c,φ(t)c),whereφ(t) satisfies equationφ(t)+φ^k-1=0.Through simple arithmetic, we can get the variational equation of system (6) along Z(t) isAs(?)(c)is Hessian matrix, and then can be diagonalizable, i.e., there exists a n×n orthogonal matrix U on C, such that under the transformation the variational equation (7) can be rewritten asehere {λ₁,…,λ_n] are the n eigenvalues of the matrix (?)(c).Morales-Ruiz [15] given an necessary condition for the completely integrability of system (6):Theorem 2 Assume that Hamiltonian system (6) is meromorphic completely integrability, i.e., it has n functional independent meromorphic first integrals in involution. Then, every pair (k,λ_i),i=1,…,n must be in one of the cases belowwhere p is arbitrary integer number, A is arbitrary complex number.Furthermore, in his Doctoral dissertation[26], Sergi Simon i Estrada obtain a malleable result of theorem 2:Theorem 3 Assume that Hamiltonian system (6) has m functional independent meromorphic first integrals in involution f₁,…,f_m on the Riemann surface determined by the particular solution Z(t). Then,1. There are at least m eigenvalues of (?)(c),without loose of generality, sayλ₁,…,λ_m, in the cases of theorem 2;2. If system (6) has another meromorphic first integral f,such that it is independent with f₁,…,f_m on a neighborhood of determined by the particular solution Z(t) (not necessary on the Riemann surface), then There are at least one of the remain eigenvaluesλ_m+1,…,λ_n in the cases of theorem 2.Consider Hamiltonian systemwhere s is a small parameter,r=(?).With theorem 3, We can obtain the meromorphic non-integrability of:Theorem 4 System (9) is not meromorphic integrable.With this result, we can correct the proof flaw in [7], and obtain the rational nonintegrability of J₂₂-problem[7].Next, we consider Hamiltonian systemwherec_i=1,…,n is positive constants,k∈Z\{0,1}.for all k,system (11) has a first integral independent with the Hamiltonian functionWith theorem 3, we have Theorem 5 whenn=3,c₁=c₂=c₃=1,k∈Z\{-2,0,1,2,4},system (10) does not have the third meromorphic first integral f₃,such that H,f₂,f₃ are functional independent.As further a discuss of the result, we consider a higher degree-of-freedom again the special situation, i.e., when n=4,k=4,c₁=c₂,c₃=c₄.We have the following result:Theorem 6 Let n=4,k=4,c₁=c₂,c₃=c₄,denote by b=c₄/c₁.Then wehnb∈R⁺\{1,(?)±(?)(?)},system (10) does not have the third meromorphic first integral f₃, such that H,f₂,f₃ are functional independent.At last, consider the following Hamiltonian systemwhere∑₁,…,i_r> represents the summations of all arrays of r different numbers in {1,…,n},andr≤n,s>1.With theorem 3, we haveTheorem 7 System(11) is not meromorphic integrable.From the theorem7, It is easy to conclude that:Corollary 1 When s=2,if n≠2r-1,and r≠1.Then system (11) is not meromorphic integrable.Then, we have answered the open problem proposed by Umeno [28] in 1995.