Ivancevic_Applied-Diff-Geom

370 Applied Differential Geometry: A Modern Introductionand G s , and where Λ G and Λ G ∗are the corresponding Lie–Poisson tensorson G and G s (see [Lu (1990)]). It is clear now that the projections µ G ∗and µ G of (D, Λ + D ) onto (G, Λ G) and (G s , Λ G ∗), respectively, are Poissonmaps. Note that we get the cotangent bundle (D, Λ + D ) = (T ∗ G, Λ 0 ) if weput Λ G = 0.The group G acts on (D, Λ + D) by left translations which, in generalare not canonical transformations. However, this is a Poisson action withrespect to the inner Poisson structure Λ G on G, which is sufficient to developthe momentum map reduction theory (see [Lu (1991)]). For our purposes,let us take a Casimir 1–form η for Λ G ∗, i.e., Λ G ∗(η) = 0. By means of themomentum mapµ G ∗ : D −→ G s , we define the vector–field on D [Alekseevsky et. al. (1997)]:Γ η = Λ + D (µs G ∗(η)).In ‘coordinates’ (g, u), due to the fact that η is a Casimir, we getΓ η (g, u) =< Y ri , η > (u)X l i(g),so that Γ η is associated with the Legendre mapL η : D ≃ G × G s −→ T G ≃ G × G,L η (g, u) =< Y ri , η > (u)X i ,which can be viewed also as a map L η : G s −→ G. Thus we get the followingTheorem [Alekseevsky et. al. (1997)]: The dynamics Γ η on the group doubleD(G, Λ G ), associated with a 1–form η which is a Casimir for the Lie–Poissonstructure Λ G ∗ on the dual group, is given by the system of equations˙u = 0,g −1 ġ =< Y ri , η > (u)X i ∈ G,and is therefore completely integrable by quadratures.We have seen that if we concentrate on the possibility of integrating oursystem by quadratures, then we can do without the requirement that thesystem is Hamiltonian.By considering again the equations of motion in action–angle variables,we classically have˙ I k = 0,˙φk = ν k (I).Clearly, if we have˙ I k = F k (I), ˙φk = A j k (I)φ j, (3.185)