Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be chosen from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.


INTRODUCTION
Let H : D → R be a smooth function where D is an open set in R 2k . The Hamiltonian system is defined by the ordinary differential equations dq j dt = ∂H ∂p j (p, q), dp j dt = − ∂H ∂q j (p, q) (j = 1, . . . , k) where (p, q) = (p 1 , . . . , p k , q 1 , . . . , q k ) ∈ D. The function H is called the Hamiltonian and k is called the degrees of freedom. A function F : D → R is called the first integral of (1) if F is conserved along each solution of (1). For two functions F, G : D → R, the Poisson bracket is the function defined by A function F : D → R is a first integral of (1) if and only if {F, H} is identically zero. Hamiltonian system (1) is called integrable if there are k first integrals F 1 (= H), F 2 , . . . , F k such that dF 1 , . . . , dF k are linearly independent in an open dense set of D and that {F i , F j } = 0 for any i, j = 1, . . . , k.
The dynamics of the integrable systems are well understood because of the Liouville-Arnold theorem(see [1,Chapter 10]) while the dynamics of the nonintegrable Hamiltonian systems may be chaotic. Therefore it is important to distinguish between integrable and non-integrable Hamiltonian systems.
This problem have been studied for quite long time. Bruns [2] proved that in the 3-body problem there is no algebraic first integral which is independent from the known ones. After that, Poincaré [4] proved that the perturbed Hamiltonian systems there is no analytic first integral depending analytically on a parameter. Then by applying it to the restricted 3-body problem, he proved the non-existence of an analytic first integral depending analytically on a mass parameter.
Another theory in this field was originated by Kovalevskaya [3]. By studying the property of singularities she discovered a new integrable case in the rigid body model. As a development of her approach, Ziglin [5,6] established the theory of singularity for proving the non-integrability. By applying the Ziglin analysis, Yoshida [7] provided a criterion for the non-integrability of the homogeneous Hamiltonian systems. Morales-Ruiz & Ramis [8,9] extended the Ziglin analysis by applying the Differential Galois theory (Picard-Vessiot theory). The Morales-Ramis theory is the strongest in this field now.
Our purpose is to prove the non-integrability of Hamiltonian systems from a new approach. We consider a Hamiltonian system of 2 degrees of freedom with a homogeneous potential of degree β ∈ R. Its Hamiltonian is represented by Here U is a real-meromorphic function on R 2 \{0} and satisfies the homogeneous property: Let V (θ) = U (cos θ, sin θ).
Above we used the word "real-meromorphic". We call a real function f (p, q) real-meromorphic if and only if f (p, q) is analytic in all but possibly a discrete subset of R 2 × (R 2 \{0}) and these exceptional points must be poles. Remark 1. The case of θ 1 = θ −1 + 2π is allowed in assumption 2. These two critical points are essentially identical. In this case, just two critical points of V are necessary.

Remark 2.
In the case of β = −2, the Hamiltonian system is integrable. Because a function is a first integral. Hence this case does not need to be studied. If V is a constant, the system is integrable. Hence we need to consider the non-constant functions. Generically there are several critical points of V and the graph is convex at some of them. The assumption 1-5 of this theorem is not strong, and only assumption 6 is a little strong. This paper is organized as follows. In Section 2 we introduce the McGehee's blowing-up technique for the homogeneous Hamiltonian systems. We prove our theorem in Section 3 by using the McGehee's technique. We present two applications of the theorem in Section 4. In the final section we compare our theorem with the Morales-Ramis theorem.

MCGEHEE'S BLOWING UP TECHNIQUE
McGehee [10] established a blowing-up technique for the triple collision singularity in the collinear three-body problem. We can easily extend the technique for the general homogeneous Hamiltonian systems (2).
We fist consider the case of β < 0. The McGehee coordinates (r, θ, v, w) are defined by Then the equations become In these coordinates the total energy is Fix the energy constant at any non-zero value(h = 0). The point q = 0 is singularity of the differential equations, but r = 0 is not singular in these differential equations (3)- (6). It is sufficient to consider the three equations (4), (5) and (6), since these equations are independent from r and since r can be obtained from (7).
The set is invariant. In the case of the n-body problem, M is called the collision manifold. Orbits converge to M as r → 0.
In the case that β > 0, we can discuss similar argument by letting R = r −1 . The equation (3) and the total energy is The equations can be extended to R = 0. Orbits converge to the invariant set M as R → 0. It is sufficient to consider the three equations (4), (5) and (6).
This means that the v-component is monotone along each solution excluding equilibrium points since all orbits on , 0) are equilibrium points of (4), (5), (6). The linearized equations of (4), (5) The eigenvalues of the coefficient matrix are λ 1 = ∓β 2V (θ c ), λ 2 and λ 3 where λ 2 and λ 3 are the roots of equation The eigenspace corresponding to λ 1 is perpendicular to M at the equilibrium point and the eigenspace corresponding to λ 2 and λ 3 is tangent to M.

PROOF OF THEOREM 1
Assume that Φ(p, q) is a real-meromorphic first integral of (2). From the homogeneous property if (p(t), q(t)) is a solution, so is (c β p(c β−2 t), c 2 q(c β−2 t)) for any constant c > 0. Then Φ(c β p, c 2 q) is also an first integral. The point (p, q) = (0, 0) may be an essential singularity of Φ. Consider the Laurent series at this point: Then we get We gather the terms according to the power of c where By substituting bc for c of (10), we get and by substituting c, p and q for b, c β p, c 2 q of (10), we get These equations (11) and (12) deduce Therefore we get f ω (c β p, c 2 q) = c ω f ω (p, q).
The function Ψ(1, θ, v, w) is real-meromorphic of (θ, v, w). Note that we do not need analyticity at r = 0 because of r = 1.
We denote the equilibrium points by We also use local coordinates (θ, w, z) near D − l where The surface M corresponds to the plane z = 0. In these coordinates, the energy is represented by Define a function g on a neighborhood by which is real-meromorphic where the coordinates work. Because Ψ is realmeromorphic, we can consider the Laurent series of g at z = 0 with respect to z: where ν is an integer and γ ν (θ, w) is not identically zero. Hence the first integral is represented by ∞ k=ν γ k (θ, w)z k =: Ξ(θ, w, z).
If Φ depends only on H, Ξ is a constant function. From here the proof varies according to ν − ρ 2β .
The case of ν − ρ 2β < 0. Take any P ∈ W s (D − 0 )\M near D − 0 . Let a = Ξ(P ). We take a small neighborhood of P is satisfied. Let ϕ τ (θ, z, w) be the flow of the differential equations. Since the first integral is conserved along each orbit, (14) holds in From the continuity, (14) also holds its closure N ε . This set N ε includes the unstable manifold is an open set of M. The z-component converges to zero as Q goes close to M. Hence γ ν must be zero on W u (D − 0 ). From the analyticity, γ ν is identically zero. This contradicts the assumption for γ ν . The case of ν − ρ 2β > 0. Consider the case of V (θ 1 ) ≤ V (θ −1 ). The other case is essentially same. Take any Q ∈ W s (D − 1 )\M. The first integral has a value c along the orbit passing Q: The z-component of ϕ τ (Q) converges to 0 as τ diverges to infinity, then c must be 0. Therefore . Because of the continuity, Ξ(Q) is zero on W s (D − 1 ). We can write the function Ξ as ∂θ 2 (θ 1 ) < 0, the equilibrium point D − 1 is hyperbolic and λ 2 λ 3 < 0. Hence there are stable and unstable manifolds with dimension 1 on M. The dynamics near the equilibrium point D − 0 is stable focus and the flow on M is gradient-like with respect to the v-component. Hence W u (D − 1 ) twins around D − 0 and Ξ is equal to zero on the spiral curve. γ ν is also zero there. Therefore from analyticity γ ν (θ, w) ≡ 0. This is a contradiction.
The case of ν − ρ 2β = 0. In this case γ ν is a first integral for the flow on M. From the similar argument as the previous case, γ ν is a constant c. Ξ − c is also a first integral. If Ξ − c is not identically zero, Ξ − c has zero point of finite degree at z = 0. This is reduced to the case of ν − ρ 2β > 0. This completes the proof for −2 < β < 0.
The proof for the other β is essentially same. We survey the cases. Consider the case of β < −2.
is an open set of M, γ ν must be a zero function.
The case of ν − ρ 2β = 0 γ ν must be constant W s/u (D − l ). If Ξ is not constant function, this case can be reduced to the case of ν − ρ 2β > 0. Finally consider the case of β > 0.
The case of ν + ρ 2β > 0 γ ν must be zero is an open set of M. γ ν must be a zero function.
The case of ν + ρ 2β = 0 γ ν must be constant W s/u (D − l ). If Ξ is not constant function, this case can be reduced to the case of ν + ρ 2β > 0.

APPLICATION
The Isosceles Three-Body Problem In the planar isosceles three-body problem, we can take the centre of gravity as the origin and the symmetric axis as the y-axis, and assume that the equal masses are located at By applying Theorem 1, we obtain: Theorem 2. If α < 55 4 , the isosceles three-body problem has no real-meromorphic first integral independent from H.
In fact, it is known that the dynamics is complex in the case of α < 55 4 . For example there are infinitely many heteroclinic orbits [11,12].

Yoshida's Example Consider the Hamiltonian
which was written on Yoshida's paper [13]. As we stated at Remark 3, we can consider the Hamiltonian instead of H. By applying Theorem 1, we obtain: Theorem 3. If ε < − 1 8 or ε > 25 7 , the Hamiltonian system (16) has no realmeromorphic first integral independent from G.

COMPARISON WITH THE MORALES-RAMIS THEORY
We call a configuration c ∈ R 2 the Darboux point of U if ∇U (c) = c. Consider the Hessian matrix of U at c and call its eigenvalues Yoshida coefficients at c. Since U is homogeneous with degree β, we can easily show that one of Yoshida coefficients is β −1. As computed by Sansaturio et al [14], the other (non-trivial) Yoshida coefficient is represented by in the polar coordinates where ∂V ∂θ (θ c ) = 0. In our theorem the assumption 6 can be written as by using λ. Then, in other words, if an integrable Hamiltonian system satisfies the assumption 1-5, the Yoshida coefficients at each Darboux point satisfy The Morales-Ramis theorem gave a list of the Yoshida coefficient which integral systems can have. We have compared the inequality (17) and the Morales-Ramis' list. The integrable list given by Morales-Ramis is included in our region (17) for β ∈ Z\{±2, 0}. For example, in the case of β = −1, from the Moreles-Raims theorem, the Yoshida coefficient of an integrable system must be in According to our theorem, the Yoshida coefficient of an integrable system must be no more than 9/8 if the other assumptions 1-5 are satisfied.
In the example of the isosceles three-body problem, the Morales-Ramis theory guarantees the non-existence of meromorphic first integral for any α. In the Yoshida's example, Morales-Ramis theory guarantees the non-existence of meromorphic first integral excluding ε = 0, 1, 3. The same result have been obtained through the Ziglin analysis [13]. It is known that these exceptional three cases are actually integrable.
We compare our theorem with the Morales-Ramis theory in several viewpoints.
Homogeneous degree Our theorem can be applied to the case of any real number β excluding −2, 0 while the result from an application [8] of Morales-Ramis theory can be apply to the case of any integer excluding β = −2, 0, 2. The case of β = −2 does not need to be studied since the systems are integrable as we stated at Remark 2. Our theorem alone can be applied to the case of β = 2 1 . Neither show anything in the case of β = 0.
Degrees of freedom Our theorem can be applied to two degrees of freedom while Morales-Ramis theory can be applied to any degrees of freedom.
Yoshida coefficients In the case of integer β except 0, ±2, the assumption which is imposed in the Morales-Ramis theory is wider than ours for proving the non-integrability.
Class of functions Our function class of first integrals is bigger. We prove the non-existence of first integral which is meromorphic as a real function in R 2 × (R 2 \{(0, 0)}), while M-R theory prove the non-existence of first integrals which is meromorphic as a complex function. Moreover only our class of functions allows essential singularities at the exceptional points: q = 0, q = ∞, p = ∞.
Proof methods Proofs are quite different. Our proof is simpler and based on dynamics (the behavior of stable and unstable manifolds). the proof of Morales-Ramis theory is far from the theory of the dynamics since that is based on the complex analysis and the differential Galois theory.