Non-complete integrability of a satellite in circular orbit (Q2492143)

Summary: We consider the problem of a rigid body (for example, a satellite) moving in a circular orbit around a fixed gravitational center whose inertia tensor components \(A,B,C\) are positive real numbers satisfying \(0<A<B\leq C=1\). We prove the non-complete meromorphic integrability of the satellite using a criterion based on a theorem of \textit{J.-J. Morales Ruiz} and \textit{J.-P. Ramis} [Methods Appl. Anal. 8, No. 1, 33--95 (2001; Zbl 1140.37352); ibid., 97--111 (2001; Zbl 1140.37354)]. This criterion relies on some local and global properties of a liner differential system, called normal variational system and depending rationally on \(A\) and \(\sqrt{3(B-A)}\). Our proof uses tools from computer algebra and proceeds in two steps: first, the satellite with axial symmetry (i.e. \(0<A<B=C=1)\), then the satellite without axial symmetry (i.e. \(0<A<B<C=1\)).