Haruo YOSHIDA

National Astronomical Observatory
Mitaka
Tokyo 181 
Japan
yoshida@gauss.mtk.nao.ac.jp
 

A new necessary condition for the integrability of Hamiltonian systems with a two dimensional homogeneous potential

Abstract: Recently, Morales-Ruiz and Ramis obtained a strong necessary condition for integrability of Hamiltonian systems with a homogeneous potential, based on their own theorem on the differential Galois theory (Picard-Vessiot theory) for Hamiltonian systems. The theorem claims that if the original Hamiltonian system is integrable, then the variational equation around a particular solution is solvable in the sense of the differential Galois theory, i.e., the solution is obtained only by a combination of quadratures, exponential of quadratures and algebraic functions. In this paper, an elementary and independent proof of this statement is given for Hamiltonian systems with a two dimensional homogeneous potential, which leads to a new necessary condition for integrability. This new necessary condition well justifies the so-called weak Painlev\'{e} conjecture of Ramani et al. for the first time.