arXiv subjects

Thierry Combot

Publications and source records attributed to Thierry Combot.

23 records · Page 2Linked to original sources

arXiv2013

Meromorphically integrable homogeneous potentials with multiple Darboux points

We prove that the only meromorphically integrable planar homogeneous potential of degree k <> -2,0,2 having a multiple Darboux point is the potential invariant by rotation. This case is a singular case of the Maciejewski-Przybylska relation on eigenvalues at Darboux points of homogeneous potentials, and needed before a case by case special analysis. The most striking application of this Theorem is the complete classification of integrable real analytic homogeneous potentials in the plane of negative degree.

math.DS↗

arXiv2012

A note on algebraic potentials and Morales-Ramis theory

We present various properties of algebraic potentials, and then prove that some Morales-Ramis theorems readily apply for such potentials even if they are not in general meromorphic potentials. This allows in particular to precise some non-integrability proofs in celestial mechanics, where the mutual distances between the bodies appear in the potentials, and thus making this analysis unavoidable.

math.DS↗

arXiv2012

Third order integrability conditions for homogeneous potentials of degree -1

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,θ)=r^{-1} h(\exp(iθ)) with h a polynomial of degree less than 3. We find then all meromorphically integrable potentials of this form.

math.DS↗

arXiv2011

Non integrability of the n body problem with non zero angular momentum

We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree -1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n body problem with Newtonian interaction in the plane on a surface of equation $(H,C)=(H_0,C_0)$ with $(H_0,C_0) \neq (0,0)$ where C is the angular momentum and H the energy, in the case where the n masses are equal.

math.DS↗

arXiv2011

Integrability conditions at order 2 for homogeneous potentials of degree -1

We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1. Conversely, we prove that if this criterion is satisfied, then the Galois group of second order variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.

math.DS↗