Journal of Applied Mathematics
Volume 1 (2001), Issue 1, Pages 1-28
doi:10.1155/S1110757X01000043
Abstract mechanical connection and abelian reconstruction for
almost Kähler manifolds
Control and Dynamical Systems, 107-81, California Institute of Technology, Pasadena 91125, CA, USA
Received 12 September 2000; Revised 20 February 2001
Copyright © 2001 Sergey Pekarsky and Jerrold E. Marsden. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.