Linearised higher variational equations
Sergi Simon
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4827-4854 doi: 10.3934/dcds.2014.34.4827
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations $\mathrm{LVE}_{\psi}^k$ of a generic autonomous system along a particular solution $\psi$. The main result of this paper is a compact yet explicit and computationally amenable form for said variational systems and their monodromy matrices. Alternatively, the same methods are useful to retrieve, and sometimes simplify, systems satisfied by the coefficients of the Taylor expansion of a formal first integral for a given dynamical system. This is done in preparation for further results within Ziglin-Morales-Ramis theory, specifically those of a constructive nature.
keywords: Zariski topology functions of several complex variables. algebraic Geometry differential Galois group variational equations monodromy matrix meromorphic functions Hamiltonian integrability multilinear algebra dynamical integrability first integrals
On the meromorphic non-integrability of some $N$-body problems
Juan J. Morales-Ruiz Sergi Simon
Discrete & Continuous Dynamical Systems - A 2009, 24(4): 1225-1273 doi: 10.3934/dcds.2009.24.1225
We present a proof of the meromorphic non--integrability of the planar $N$-Body Problem for some special cases. A simpler proof is added to those already existing for the Three-Body Problem with arbitrary masses. The $N$-Body Problem with equal masses is also proven non-integrable. Furthermore, a new general result on additional integrals is obtained which, applied to these specific cases, proves the non-existence of an additional integral for the general Three-Body Problem, and provides for an upper bound on the amount of additional integrals for the equal-mass Problem for $N=4,5,6$. These results appear to qualify differential Galois theory, and especially a new incipient theory stemming from it, as an amenable setting for the detection of obstructions to Hamiltonian integrability.
keywords: Obstructions to integrability (nonintegrability criteria) in Hamiltonian systems $N$-body problems. Dynamical systems in classical and celestial mechanics differential algebra

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