Seminar

Geometry, Mechanics and Control Seminar

Integrability by quadratures of the Lie group`s exponential curves

Speaker:  Sergio Grillo (Instituto Balseiro, Universidad Nacional de Cuyo and CONICET, Argentina)
Date:  Friday, 28 May 2021 - 15:30
Place:  Online - zoom.us/j/96739216821?pwd=UDhPYlNzM0M3dXFHUnJRSW92enl1dz09

Abstract:
In this talk we present an integrability by quadratures theorem valid for a class of dynamical systems that generalize the non-commutative integrable ones. Such a class is given by systems defined on a symplectic manifold, but whose related vector fields do not need to be Hamiltonian. Then we apply the theorem to certain dynamical systems defined on the cotangent bundle of a Lie group G, whose integral curves are given, in essence, by exponential curves of G. This enable us, for some elements ξ of the Lie algebra g of G, to write down an explicit expression up to quadratures of the curves exp ( ξ t ), different from that appearing in the literature for matrix groups. In the case of compact and of semisimple Lie groups, we show that such expression of exp ( ξ t ) is valid for all ξ inside an open dense subset of g.