Ivancevic_Applied-Diff-Geom

368 Applied Differential Geometry: A Modern Introductionwith the (trivial) natural Poisson structure of the dual of a Lie algebra.It is now clear that any function on (R n ) s , when pulled back to T ∗ R n orT ∗ T n , induces a Hamiltonian system which is completely integrable (inthe Liouville sense). Because the level sets of this function carry on theaction of R n , the completely integrable system induces a 1D subgroup ofthe action of R n on the given level set. However, the specific subgroup willdepend on the particular level set, i.e., the ‘frequencies’ are first integrals.The property of being integrable by quadratures is captured by the factthat it is a subgroup of the R n −action on each level set.It is now clear, how we can preserve this property, while giving up therequirement that our system is Hamiltonian. We can indeed consider any1–form η on (R n ) s and pull it back to T ∗ R n or T ∗ T n , then associatedvector–field Γ η = Λ 0 (µ s (η)), where Λ 0 is the canonical Poisson structurein the cotangent bundle, is no more Hamiltonian, but it is still integrableby quadratures. In action–angle variables, if η = ν k dI k is the 1–form on(R n ) s , the associated equations of motion on T ∗ T n will be [Alekseevskyet. al. (1997)]˙ I k = 0, ˙φk = ν k ,with ˙ν k = 0, therefore the flow will be as in (3.183), even though ∂ I j ν k ≠∂ I kν j .We can now generalize this construction to any Lie group G. We considerthe Hamiltonian action G × T ∗ G → T ∗ G, of G on the cotangentbundle, induced by the right action of G on itself. The associated momentummap µ : T ∗ G ≃ G s × G −→ G s . It is a Poisson map with respect tothe natural Poisson structure on G s (see, e.g., [Alekseevsky et. al. (1994);Libermann and Marle (1987)]).Now, we consider any differential 1–form η on G s which is annihilatedby the natural Poisson structure Λ G ∗ on G s associated with the Liebracket. Such form we call a Casimir form. We define the vector–fieldΓ η = Λ 0 (µ s (η)). Then, the corresponding dynamical system can be writtenas [Alekseevsky et. al. (1997)]g −1 ġ = η(g, p) = η(p), ṗ = 0,since ω 0 = d(< p, g −1 dg >) (see [Alekseevsky et. al. (1994)]). Here weinterpret the covector η(p) on G s as a vector of G. Again, our system canbe integrated by quadratures, because on each level set, get by fixing p’sin G s , our dynamical system coincides with a one–parameter group of the