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

Abstract

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 variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.

Third order integrability conditions for homogeneous potentials of degree −1

“…This theorem is in fact a particular case of Theorem 2 in [5] for which the three indices i, j, k are equal.…”

Integrability and Non Integrability of Some n Body Problems

“…To study the Galois group of second order variational equations, we apply Theorem 2 of [17]. We have however to take into account that the kinetic energy is p 2…”

“…According to [17], the condition for integrability of the second order variational equations are that some of these third order derivative should vanish. Using the table of [17], the three first third order derivatives never lead to an integrability condition, but the last one does. In particular, for k = 5, 14, the integrability condition is D 3 V (X 3 , X 3 , X 3 ) = 0, and there are none for k = 9.…”

“…k-th variational equations is put under block triangular form to make computation faster. Only the last equation is solved through the variation of parameters technique and then its monodromy analyzed through commutativity condition of monodromy in [17]. Instead of computing a basis of solutions, we only compute several solutions, that through empirical evidence, will lead to the strongest integrability conditions.…”

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

“…In particular, they have zero "norm". But if W has only simple eigenvalues, then W is diagonalizable and using Theorem 6 of [11], W is then diagonalizable in an "orthonormal" basis. So if W has an eigenvector with zero "norm", then this eigenvector is a linear combination of two eigenvectors and this implies an eigenspace of dimension greater than 2 and then a double eigenvalue.…”