Linearizing Completely Integrable Systems on Complex Algebraic Tori
How does one recognize whether a Hamiltonian system is completely integrable ? Granted a system is integrable, how does one effectively integrate the problem ? The answer to these questions is unknown, up to this day! The proof of the Liouville theorem concerning integrals in involution and invariant tori is non-constructive : neither does it enable you to decide about its integrability, nor does it provide means for integrating the problem. Historically, the solutions to the classical systems that we know have required the most ingenious tricks by the best mathematical minds and they are only distinguished by their variety. The resolution of the Korteweg-de Vries equation by inverse spectral methods, some twenty years ago, has led to a number of Lie theoretical and algebraic geometrical methods for producing ordinary and partial differential equations, some of which are interesting from the point of view of mechanics and physics. They all give rise to so-called algebraically completely integrable systems; this stringent, but yet quite typical notion of integrability will be defined in § 2, whereas § 1 deals with the simple but fundamental example of Euler’s rigid body rotating about its center of mass.
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