Hamiltonian dynamics on matched pairs

Abstract

The cotangent bundle of a matched pair Lie group, and its trivialization, are shown to be a matched pair Lie group. The explicit matched pair decomposition on the trivialized bundle is presented. On the trivialized space, the canonical symplectic two-form and the canonical Poisson bracket are explicitly written. Various symplectic and Poisson reductions are perfomed. The Lie-Poisson bracket is derived. As an example, Lie-Poisson equations on sl(2, C) * are obtained.

Hamiltonian coupling of electromagnetic field and matter

“…In this section, we start with two Lie groups G and K under the mutual interactions. Recall that on the cotangent bundle of a Lie group we have a canonical Poisson bracket (20), and in the reduced picture (that is on the dual space g * ) there exists Lie-Poisson bracket (17). In this section, we present the most general way to couple (match) two canonical Poisson brackets as well as to couple (match) two Lie-Poisson brackets.…”

“…Being a cotangent bundle, T * (G ⊲⊳ K) is a symplectic manifold hence it is equipped with the canonical Poisson bracket {•, •} T * (G⊲⊳K) . Although it has technically the same structure with the canonical Poisson bracket {•, •} T * G on T * G presented in (20), this time the trivialization (19) enables us to recast the canonical Poisson bracket on T * (G ⊲⊳ K) in terms of actions and the canonical Poisson structures on T * G and T * K.…”

“…where F g is an element of T * g G, T * R is the cotangent lift of the right translation map on the group level, H ν ⊲ g is the infinitesimal action of k on G, F ν ⊳ g is the lift of the action of G on k, and F µ is an element of g, [20]. Note that while presenting the matched canonical Poisson bracket (39), we have combined some of the terms and labeled them.…”

“…Note that while presenting the matched canonical Poisson bracket (39), we have combined some of the terms and labeled them. The first line is the canonical Poisson bracket on T * g G also given in (20). If the function(al)s depend only on g and ν in the expression of the matched canonical Poisson bracket (39), remains the second line remains, which is the Poisson bracket on k * ⋊ G. As we shall point out later on, existence of this reduced Poisson bracket enables one to write magnetohydrodynamics and electrohydrodynamics as two subsystems of the electromagnetohydrodynamics in a geometric way.…”

“…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.…”