Integrability of Hamiltonian systems and differential Galois groups of higher variational equations

Abstract

Given a complex analytical Hamiltonian system, we prove that a necessary condition for meromorphic complete integrability is that the identity component of the Galois group of each variational equation of arbitrary order along each integral curve must be commutative. This was conjectured by the first author based on a suggestion made by the third author due to numerical and analytical evidences concerning higher order variational equations. This non-integrability criterion extends to higher orders a non-integrability criterion (Morales-Ramis criterion), using only the first order variational equation, obtained by the first and the second author. Using our result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped to Morales-Ramis criterion.

Second-order variational equations forN-body simulations

“…We follow the work of Morales-Ruiz et al (2007) and start with an analytic differential equation of the forṁ…”

“…In this paper, we will only use variational equations up to second order (see e.g. Morales-Ruiz et al 2007, for equations up to order 3).…”

Periodic orbits and non-integrability in a cosmological scalar field

“…The monodromy matrix associated to the T -periodic solution φ(t, x 0 ) is the solution M (T, x 0 ) of (20) satisfying that M (0, x 0 ) is the identity matrix. The eigenvalues λ of the monodromy matrix associated to the periodic solution φ(t, x 0 ) are called the multipliers of the periodic orbit.…”

“…In fact, in order to prove the nonexistence of an additional meromorphic first integral on a non-zero Hamiltonian level the authors applied the Morales-Ramis theory (see [19] or [20]). Notice that the nonintegrability in this case is based in the non-existence of any additional meromorphic first integral in the sense of Liouville-Arnold.…”

Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems

“…The effective calculation of the Stokes multipliers of r F pxq (= the nontrivial entries of the Stokes-Ramis matrices associated with r Y pxq, see Definition 5.1) is crucial in a large number of theoretical and practical problems (calculation of differential Galois groups [24,25], integrability of some Hamiltonian systems [26,27], etc.). Thereby, in the last decades of the twentieth century, several approaches, issuing from the summability and multisummability theories and essentially based on integral methods such that Cauchy-Heine integral and Laplace transformations, were given by many authors under more or less generic assumptions on system (A) (see for instance [2,4,6,7,8,9,11,12,15,20]).…”