Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

Abstract

When the phase space P of a Hamiltonian G-system (P, ω, G, J, H) has an almost Kähler structure, a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form A are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (due to the work of A. Blaom), thus significantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.

Reconstruction phases in the planar three- and four-vortex problems

“…and I n (λ), n = 0, 1, 2 are the three roots of the cubic polynomial in (36) for which Aref and Pomphrey give explicit expressions [3]. From the theory of elliptic functions and integrals, (37) and (38) In general, some integral multiple of these time periods will be the time period T of the periodic orbit on the symplectic reduced space-the (I 1 , φ 1 ) space. Note that I 1 is proportional to the oriented area of the vortex triangle but I := (I 1 /I 2 ) 2 masks orientation information.…”

“…From the theory of elliptic functions and integrals, (37) and ( 38) are periodic with period equal to twice the complete elliptic integral of the first kind K(m), where m is the modulus of the elliptic function. Transforming to time t, the time period of the I variable is obtained as In general, some integral multiple of these time periods will be the time period T of the periodic orbit on the symplectic reduced space-the (I 1 , φ 1 ) space.…”

“…A noteworthy exception is Montgomery's work on phases in the dynamics of a free rigid body where, in addition to a geometric phase formula, a simple exact formula for the dynamic phase is obtained; see [32]. In the framework of Poisson reduction and in the framework of almost Kähler manifolds, respectively, Blaom [6] and Pekarsky and Marsden [38] have also obtained some general formulas for dynamic and geometric phases.…”

Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction

“…We mention this as a possible link to the notion of the abstract mechanical connection which is a Lie-algebra valued connection one-form deÿned on the phase space of a system with symmetry. This subject is explored in detail in the papers of Blaom (2000) and Pekarsky and Marsden (2001).…”