Normalization of Resonant Hamiltonians

Abstract

The definition of integrability is simple to state; an autonomous N degree of freedom Hamiltonian is integrable if N independent global invariants exist and these are in involution with each other.1 However, a failure to find such a set of global invariants does not exclude the possibility that the Hamiltonian in question is integrable. The detection of integrability is thus a critical issue in non-linear dynamics and a variety of analytical and numerical procedures has been developed to determine if a Hamiltonian is integrable. The most obvious approach is to try to establish if the Hamiltonian is separable, possibly using the Stäckel conditions to guide one to appropriate coordinate system. It should be noted, however, that finding coordinates that separate a particular problem can itself be a difficult task. More general and systematic approaches than simply seeking separability are therefore in order, e.g., the Whittaker program. An alternative method is the Painlevé test2 in which the analytic structure of the equations of motion in the complex time plane is examined. This approach has been used to uncover integrability but must be applied gingerly because it cannot be guaranteed to succeed in every case. Probably the simplest numerical method is to generate Poincaré surfaces of section and determine by eye whether or not the motion is integrable. Of course, no numerical method by itself can definitively determine integrability.