We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r,\theta)=r^{-1} h(\exp(i\theta)) with h a polynomial of degree less than 3. We find then all meromorphically integrable potentials of this form.
In this article, we will be interested in non-integrability proofs of meromorphic homogeneous potentials of degree -1 in the plane, and in particular in nongeneric cases. Writing our potential V in polar coordinates, and making the Fourier expansion in the angle gives us
This type of potential covers many physical problems in celestial mechanics and n-body problems, in particular the anisotropic Kepler problem, the isosceles 3-body problem, the colinear 3-body problem, the symmetric 4-body problem and so on. Moreover, for such a potential there are strong integrability conditions, thanks to the Morales-Ramis theory 1 and to a very effective criterion of Yoshida 2 . Still, for such a general potential, this criterion will not be sufficient. This is not particularly because this class of potentials is large, but because there are nongeneric, very resistant cases inside. For example, if we want to study the integrability of V (r, θ) = r -1 h(exp(iθ)) with a polynomial h, we have a priori a potential with deg h + 1 complex parameters, and Yoshida’s integrability criterion will restrict this family to a family with deg h -1 integer parameters. Still one would like to have a finite list of possible integrable potentials, so as to be able to check the existence of first integrals one by one. Here we will present a stronger criterion in Theorems 2 and 3 which is able to deal with such families, and which therefore is capable to settle any integrability question on finite dimensional families of type (1). As an application of our method, we will apply this criterion in the case V (r, θ) = r -1 h(exp(iθ)) with h ∈ C[z], deg h ≤ 3. To do precise statements, let us now begin with some definitions concerning homogeneous potentials and integrability.
Definition 1. We consider the algebraic variety S = {(q 1 , q 2 , r) ∈ C 3 , r 2 = q 2 1 + q 2 2 } and the derivations for a function f on S
where ∂ i is the derivative according to the i-th variable (the variables of f are q 1 , q 2 , r in this order). This defines a symplectic form on C 2 × S on which we consider a Hamiltonian H of the form H(p 1 , p 2 , q 1 , q 2 , r) = 1 2 (p 2 1 + p 2 2 ) -V (q 1 , q 2 , r) with the associated system of differential equations ṙ = r -1 (q 1 q1 + q 2 q2 ), qi =
The potential V is assumed to be meromorphic on S and to have the following form in polar coordinates:
This implies that V is homogeneous of degree -1. We say that I is a meromorphic first integral of H, if I is a meromorphic function on C 2 × S such that
Obviously, the Hamiltonian H itself is a first integral. We will say that V is meromorphically integrable if it possesses an additional meromorphic first integral which is independent almost everywhere from H.
where α ∈ C is called the multiplicator. Because V has singularity at c 3 = 0, we will always assume that c 3 = 0. Because of homogeneity, we can always choose α = 0 or α = -1. We say that c is non-degenerate if α = 0. To the Darboux point c we associate a homothetic orbit given by
with φ satisfying the following differential equation
In the following, we will often omit the last component of a Darboux point c ∈ S as it is defined up to a sign (and the choice of sign does not matter) by the two first components.
where ∇ 2 V (c) is the Hessian of V (according to derivations in q). After diagonalization (if possible) and the change of variable φ(t) -→ t, the equation simplifies to
where the λ i are the eigenvalues of the Hessian of V evaluated at the Darboux point c, i.e.,
Sp ∇ 2 V (c) ⊂ {0} .
In fact, this is not exactly the same statement as the original theorem because we allow r to appear in the potential and in the first integrals.
Proof. Let Γ ⊂ C 2 × S denote the curve defined by equation ( 4) without the singular point (q 1 , q 2 , r) = (0, 0, 0), and M an open neighbourhood of Γ in C 2 ×S such that H is holomorphic on M . The Hamiltonian H is then well defined and holomorphic on a symplectic manifold M and the additional first integral is meromorphic on M . Hence, using the main theorem of 3 , the neutral component of the Galois group of the variational equation near Γ is abelian at all orders over the base field of meromorphic functions on Γ. The variational equation is a hypergeometric equation. In 6 , Kimura classifies Galois groups of hypergeometric equations over the base field C(t). We can use this classification as the Galois group over the base field C(t) is the same as over the base field of meromorphic functions because the hypergeometric equation is a Fuchsian equation (see page 73 of 1 ). This produces the condition on the spectrum of ∇ 2 V (c). The case of a degenerate Darboux point leads to the variational
which is a Bessel equation (after a change of variables). Its Galois group over the field of meromorphic functions in t has not an abelian identity component except if λ = 0.
Note that in the case of a degenerate Darboux point, we explicitly need that the first integral is meromorphic i
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