Seminario Geometría, Mecánica y Control
Symmetric time-dependent Hamiltonian systems and affine coadjoint orbits
Ponente: Juan Carlos Marrero (ULL)Fecha: jueves 28 de mayo de 2026 - 12:00Lugar: Aula Naranja, ICMAT
Resumen:
It is well-known that the phase bundle of a time-dependent Hamiltonian system is a principal line bundle, whose total space is the cotangent bundle \(T^*Q\) of the total space \(Q\) of the configuration bundle endowed with the canonical symplectic structure. This symplectic structure is invariant under the principal action and the time-dependent Hamiltonian dynamics is an evolution vector eld on the base space of the principal bundle.
In this talk, I will focus in the particular case when \(Q\) is a Lie group \(G\) and the cotangent lift of the left action of \(G\) on itself and the principal \(R\)-action on \(T^*G\) commute. Then, we will see that \(G\) admits a normal subgroup \(H\) of codimension 1 and a suitable Marsden-Weinstein symplectic reduction by the symmetry Lie group H produces a reduced phase bundle, which is again a symplectic principal \(R\)-bundle. We will give a description of the total space of the bundle as an affine coadjoint orbit of an extension of \(G\) and we will see that the base space admits a Poisson structure of corank 1 whose symplectic leaves are isomorphic to a fixed standard coadjoint orbit of \(H\).
The previous facts provide a good motivation to extend the results in this talk for the more general case when \(Q\) is only the total space of a principal \(G\)-bundle and the \(R\)-action and the cotangent lift of the principal action on \(T^*Q\) commute. This extension is contained in a paper in progress with C Benitez, D Iglesias and E Padron (ULL, Spain).
