Journal of Applied Mathematics

Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds

Sergey Pekarsky and Jerrold E. Marsden

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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.

Article information

J. Appl. Math., Volume 1, Number 1 (2001), 1-28.

First available in Project Euclid: 24 March 2003

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Zentralblatt MATH identifier

Primary: 37J15: Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20] 53D20: Momentum maps; symplectic reduction
Secondary: 32Q15: Kähler manifolds


Pekarsky, Sergey; Marsden, Jerrold E. Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds. J. Appl. Math. 1 (2001), no. 1, 1--28. doi:10.1155/S1110757X01000043.

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  • M. Adams and T. Ratiu, The three-point vortex problem: commutative and noncommutative integrability, Hamiltonian Dynamical Systems (Boulder, CO, 1987), Amer. Math. Soc., Rhode Island, 1988, pp. 245–257.
  • M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction, Phys. D 123 (1998), no. 1-4, 271–290.
  • A. D. Blaom, Reconstruction phases via Poisson reduction, Differential Geom. Appl. 12 (2000), no. 3, 231–252. \CMP1+764+3311 764 331
  • ––––, A geometric setting for Hamlitonian perturbation theory, to appear, 2001.
  • P. Heinzner and F. Loose, Reduction of complex Hamiltonian ${G}$-spaces, Geom. Funct. Anal. 4 (1994), no. 3, 288–297.
  • S. Kobayashi and K. Nomizu, Foundations Of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York, 1963.
  • G. G. Luther, M. S. Alber, J. E. Marsden, and J. M. Robbins, Geometric nonlinear theory of quasi-phase-matching, J. Optical Soc. Am. B 17 (2000), 932–941.
  • J. Marsden, R. Montgomery, and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc. 88 (1990), no. 436, 110.
  • J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: a Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994.
  • ––––, Mechanics and Symmetry: Reduction Theory, Texts in Applied Mathematics, Springer, 1999.
  • J. E. Marsden, T. S. Ratiu, and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys. 41 (2000), no. 6, 3379–3429. \CMP1+768+6271 768 627
  • M. Otto, A reduction scheme for phase spaces with almost Kähler symmetry. Regularity results for momentum level sets, J. Geom. Phys. 4 (1987), no. 2, 101–118.
  • A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557.