Action-angle Variables for Generic 1D Mechanical Systems
Abstract
We consider a 1D mechanical system H( P, Q)= P2+ G( Q) in action-angle variable ( P, Q) where G is a 2π-periodic analytic function with non degenerate critical points. Then, we consider a small analytic perturbation of H of the form H*( P, Q; P) = P2+ G( Q)+ η F ( P, Q; P)=: P2 + G*( P, Q; P)\,, η 1\ , where the perturbed potential G* may depend on the action P and also on parameters P ("the adiabatic actions"); indeed, this is the form of a finite dimensional mechanical system close to an exact simple resonance after averaging over fast angles and disregarding the exponentially small remainder, see [5]. Up to a finite number of separatrices and elliptic/hyperbolic points the phase space of H* is divided into a finite number of open connected components foliated by invariant circles. On every connected component we perform a (Arnold-Liouville) symplectic action-angle transformation which integrates the system. We give a complete and quantitative description of the analyticity properties of such integrating transformations, estimating, in particular, how such transformations differ from the integrating transformation for H; compare Theorem 6.1 below.