Nonintegrability of the three-body problems for the classical helium atom Available to Purchase

We give a proof of the nonintegrability of an important three-body problem in atomic physics. We consider the classical model for the helium atom in full dimension, thus completing our previous proof for the frozen planetary approximation. To our knowledge there is not any such a proof in the literature. We apply a theorem due to Morales-Ruiz and Ramis: if a Hamiltonian system, derived from a homogeneous potential is integrable, then all integrability factors, related to the Hessian of the homogeneous potentials, satisfy certain conditions related to the degree of homogeneity. In the helium atom case, these coefficients should all be discrete. We exhibit a set of nondiscrete values determined analytically. This implies the nonintegrability of the helium atom without any computer aid. We also extend this theorem to various two-electron atoms. In the case of strange helium atoms we provide a computer aided proof of nonintegrability.