Differential Galois Approach to the Non-integrability of the Heavy Top Problem

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We study integrability of the Euler-Poisson equations describing the motion of a rigid body with one fixed point in a constant gravity field. Using the Morales-Ramis theory and tools of differential algebra we prove that a symmetric heavy top is integrable only in the classical cases of Euler, Lagrange, and Kovalevskaya and is partially integrable only in the Goryachev-Chaplygin case. Our proof is alternative to that given by Ziglin (Funktsional. Anal. i Prilozhen., 17(1):8--23, 1983; Funktsional. Anal. i Prilozhen., 31(1):3--11, 95, 1997).