Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 367and we can use it as the generating function of a canonical transformation(p i , q i ) → (I i , ϕ i ):p i = ∂S∂q i ,ϕi = ∂S∂I i.A Universal Model for Completely Integrable SystemsA Hamiltonian system on a 2nD symplectic manifold M is said to becompletely integrable if it has n first integrals in involution, which are functionallyindependent on some open dense submanifold of M. This definitionof a completely integrable system is usually found, with some minor variants,in any modern text on symplectic mechanics [Arnold (1989); Abrahamand Marsden (1978); Libermann and Marle (1987); Marmo et. al. (1995);Thirring (1979)].Starting with this definition, one uses the so–called Liouville–ArnoldTheorem to introduce action–angle variables and write the Hamiltoniansystem in the form˙ I k = 0,˙φk = ∂H∂I k= ν k (I),where k ∈ {1, . . . , n}. The corresponding flow is given byI k (t) = I k (0), φ k (t) = φ k (0) + ν k t. (3.183)The main interest in completely integrable systems relies on the fact thatthey can be integrated by quadratures [Arnold (1989)].However, it is clear that even if ν k dI k is not an exact (or even aclosed) 1–form, as long as ˙ν k = 0, the system can always be integrated byquadratures.If we consider the Abelian Lie group R n , we can construct a Hamiltonianaction of R n on T ∗ R n induced by the group addition: R n × T ∗ R n →T ∗ R n . This can be generalized to the Hamiltonian action [Alekseevskyet. al. (1997)]R n × T ∗ (R k × T n−k ) → T ∗ (R k × T n−k ),of R n , where T m stands for the mD torus, and reduces to R n × T ∗ T n orT n × T ∗ T n , when k = 0.By using the standard symplectic structure on T ∗ R n , we find the momentummap µ : T ∗ R n −→ (R n ) s , (q, p) ↦→ p, induced by the naturalaction of R n on itself via translations, which is a Poisson map if (R n ) s is