Lagrangian dynamics on matched pairs

Abstract

Abstract. Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the EulerLagrange equations on the trivialized matched pair of tangent groups, as well as the EulerPoincaré equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group SL(2, C).

Hamiltonian coupling of electromagnetic field and matter

“…Recall that we have presented the cotangent bundle of a Lie group by the product of the group and the dual space in (19). By following the same understanding, we now identify the cotangent bundle T * (G ⊲⊳ K) with its right trivialization…”

“…The Lie-group characterization of the configuration spaces of the systems is imperative here to define the mutual actions. The geometrical construction we propose does not have any particular restrictions, and it can be used for any two systems in mutual interaction satisfying certain compatibility conditions [19,20]. The matched pair concept that we shall present is the most general geometric way of coupling two systems in mutual interactions.…”

Hamiltonian dynamics on matched pairs

“…In this subsection we recall the basics on the matched pairs of Lie groups, Lie algebras and Lie coalgebras. We refer the reader to [6,13,14,15,16,29,34,37] for further details on the subject.…”

“…A matched pair Lie group G ⊲⊳ H is a Lie group by itself containing G and H as two non-intersecting Lie subgroups acting on each other subject to certain compatibility conditions [13,15,14,16,34]. In our previous work [6], we have studied the Lagrangian dynamics on the matched pair Lie algebra g ⊲⊳ h, and the tangent bundle T (G ⊲⊳ H). In the present work, we investigate the geometry of the cotangent bundle T * (G ⊲⊳ H), and present the symplectic structure, together with the Hamiltonian dynamics on it.…”

“…The forth, and the last, section is reserved to study the Lie-Poisson equations (1.1) on a concrete example. Just as we did in our previous work [6], we consider SL(2, C) and its Iwasawa decomposition.…”