Completely integrable Hamiltonian systems
Integrable Hamiltonian systems are nonlinear ordinary differential equations described by a Hamiltonian function and possessing sufficiently many independent constants of motion in involution. The regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following purely differential-geometric fact: a compact and connected \(n\)-dimensional manifold on which there exist \(n\) vector fields which commute and are independent at every point is diffeomorphic to an \(n\)-dimensional real torus and each vector field will define a linear flow there. The connection with the concept of completely integrable systems is studied and applied to solve several classical problems. This work consists of the following parts: introduction; the Arnold-Liouville theorem; rigid body rotation around a fixed point, the Euler case, the Lagrange case, the Kovalevskaya case; the motion of a rigid body in ideal fluid, the Clebsch case, the Lyapunov-Steklov case; geodesic flow on the group \(\mathrm{SO}(4)\); hierarchy of integrable systems.
scientific article; zbMATH DE number 5956222
scientific article