Abstract:
We discuss the holomorphic properties of the complex continuation of the classical
Arnol’d – Liouville action-angle variables for real analytic 1 degree-of-freedom Hamiltonian
systems depending on external parameters in suitable Generic Standard Form, with particular
regard to the behaviour near separatrices. In particular, we show that near separatrices the
actions, regarded as functions of the energy, have a special universal representation in terms
of affine functions of the logarithm with coefficients analytic functions. Then, we study the
analyticity radii of the action-angle variables in arbitrary neighborhoods of separatrices and
describe their behaviour in terms of a (suitably rescaled) distance from separatrices. Finally, we
investigate the convexity of the energy functions (defined as the inverse of the action functions)
near separatrices, and prove that, in particular cases (in the outer regions outside the main
separatrix, and in the case the potential is close to a cosine), the convexity is strictly defined,
while in general it can be shown that inside separatrices there are inflection points.
Keywords:Hamiltonian systems, action-angle variables, Arnol’d – Liouville integrable systems,
complex extensions of symplectic variables, KAM theory.
References