Non complete integrability of a satellite in circular orbit

Delphine Boucher

Abstract: 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's 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 J.-J. Morales and J.-P. Ramis. This criterion relies on some local and global properties of a linear 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 {\em with axial symmetry} (i.e. $0<A<B=C=1$) then the satellite {\em without axial symmetry} (i.e. $0<A<B<C=1$).

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