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### Open Access Journals

DCDS

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.

DCDS

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.

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