Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 371and we are able to integrate the first equation by quadratures, we againhave the possibility to integrate by quadratures the system (3.185), if onlythe matrices (A j k(I(t))) commute [Alekseevsky et. al. (1997)]:(∫ t)φ(t) = exp A(I(s))ds φ 0 .0Because φ k are discontinues functions on the torus, we have to be morecareful here. However, we show how this idea works for double groups. Inthe case when the 1–form η on G s is not a Casimir 1–form for the Lie–Poisson structure Λ G ∗, we get, in view of (3.184),Γ η (g, u) =< Y ri , η > (u)X l i(g) + Λ G ∗(η)(u).Now, the momenta evolve according to the dynamics Λ G ∗(η) on G s (whichcan be interpreted, as we will see later, as being associated with an interactionof the system with an external field) and ‘control’ the evolutionof the field of velocities on G (being left–invariant for a fixed time) by a‘variation of constants’. Let us summarize our observations in the followingTheorem [Alekseevsky et. al. (1997)]: The vector–field Γ η on the doublegroup D(G, Λ G ), associated with a 1–form η on G s , defines the followingdynamics˙u = Λ G ∗(η)(u),g −1 ġ =< Y ri , α > (u)X i ∈ G, (3.186)and is therefore completely integrable, if only we are able to integrate theequation (3.186) and < Yi r,η > (u(t))X i lie in a commutative subalgebraof G for all t.Finally, we can weaken the assumptions of the previous Theorem. It issufficient to assume [Alekseevsky et. al. (1997)] thatg −1 ġ(t) = exp(tX)A(t) exp(−tX),for some A(t), X ∈ G, such that X + A(t) lie in a commutative subalgebraof G for all t (e.g., A(t) = const), to assure that (3.186) is integrable byquadratures. Indeed, in the new variablethe equation (3.186) readsg 1 (t) = exp(−tX)g(t) exp(tX),ġ 1 (t) = g 1 (t)(X + A(t)) − Xg 1 (t),