Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 369action of G on that particular level set.We give a familiar example: the rigid rotator and its generalizations[Alekseevsky et. al. (1997)]. In the case of G = SO(3) the (right) momentummapµ : T ∗ SO(3) −→ so(3) ∗is a Poisson map onto so(3) ∗ with the linear Poisson structureΛ so(3) ∗ = ε ijk p i ∂ pj ⊗ ∂ pk .Casimir 1–forms for Λ so(3) ∗ read η = F dH 0 , where H 0 = ∑ p 2 i /2 is the‘free Hamiltonian’ and F = F (p) is an arbitrary function. Clearly, F dH 0 isnot a closed form in general, but (p i ) are first integrals for the dynamicalsystem Γ η = Λ 0 (µ s (η)). It is easy to see thatΓ η = F (p)Γ 0 = F (p)p i ̂Xi ,where ̂X i are left–invariant vector–fields on SO(3), corresponding to thebasis (X i ) of so(3) identified with (dp i ). Here we used the identificationT ∗ SO(3) ≃ SO(3)×so(3) ∗ given by the momentum map µ. In other words,the dynamics is given byṗ i = 0,g −1 ġ = F (p)p i X i ∈ so(3),and it is completely integrable, since it reduces to left–invariant dynamicson SO(3) for every value of p. We recognize the usual isotropic rigid rotator,when F (p) = 1.We can generalize our construction once more, replacing the cotangentbundle T ∗ G by its deformation, namely a group double D(G, Λ G ) associatedwith a Lie–Poisson structure Λ G on G (see e.g., [Lu (1990)]). This double,denoted simply by D, carry on a natural Poisson tensor–field Λ + D whichis non–degenerate on the open–dense subset D + = G · G s ∩ G s · G of D(here G s ⊂ D is the dual group of G with respect to Λ G ). We refer to Das being complete if D + = D. Identifying D with G × G s if D is complete(or D + with an open submanifold of G × G s in general case; we assumecompleteness for simplicity) via the group product, we can write Λ + Din‘coordinates’ (g, u) ∈ G × G s in the form [Alekseevsky et. al. (1997)]Λ + D (g, u) = Λ G(g) + Λ G ∗(u) − X l i(g) ∧ Y ri (u), (3.184)where Xil and Y ir are, respectively, the left– and right–invariant vector–fields on G and G s relative to dual bases X i and Y i in the Lie algebras G