Averaging, symplectic reduction, and central extensions. (English) Zbl 1473.37069
Summary: We show that the averaged equation for a one-frequency fast-oscillating Hamiltonian system is the result of symplectic reduction of a certain natural system on the corresponding \(S^1\)-bundle with respect to the circle action. Furthermore, if the reduced configuration space happens to be a group, then under natural assumptions the averaged system turns out to be the Euler equation on a central extension of that group. This gives a new explanation of the drift, common in averaged system, as a similar shift is typically present in symplectic reductions and central extensions.
MSC:
| 37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 53D20 | Momentum maps; symplectic reduction |
Keywords:
Hamiltonian system; symplectic reduction; central extension; one-frequency system; averaged system; Euler equationReferences:
| [1] | Arnold V I 1989 Mathematical Methods of Classical Mechanics (New York: Springer) · Zbl 0692.70003 · doi:10.1007/978-1-4757-2063-1 |
| [2] | Arnold V I, Kozlov V V and Neishtadt A I 2006 Mathematical Aspects of Classical and Celestial Mechanics (New York: Springer) · Zbl 1105.70002 · doi:10.1007/978-3-540-48926-9 |
| [3] | Cox G and Levi M 2017 Magnetic terms in averaged Hamiltonian systems (arXiv:1707.04970) |
| [4] | Cox G and Levi M 2018 Gaussian curvature and gyroscopes Commun. Pure Appl. Math.71 938-52 · Zbl 1422.70004 · doi:10.1002/cpa.21731 |
| [5] | Craik D D and Leibovich S 1976 A rational model for Langmuir circulations J. Fluid Mech.73 401-26 · Zbl 0324.76014 · doi:10.1017/S0022112076001420 |
| [6] | Dellar P J 2011 Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton’s principle on a sphere J. Fluid Mech.674 174-95 · Zbl 1241.76428 · doi:10.1017/S0022112010006464 |
| [7] | Kapitza P L 1951 Dynamic stability of a pendulum when its point of suspension vibrates Sov. Phys.—JETP21 588-97 |
| [8] | Khesin B A and Chekanov Yu V 1989 Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in D dimensions Physica D 40 119-31 · Zbl 0820.58019 · doi:10.1016/0167-2789(89)90030-4 |
| [9] | Klein F 1926 Vorlesungen Über Höhere Geometrie (Berlin: Springer) 3-d Aufl. · JFM 52.0624.09 |
| [10] | Landau L D and Lifshitz E M 1979 Mechanics (Oxford: Pergamon) |
| [11] | Marsden J E, Misiolek G, Ortega J, Perlmutter M and Ratiu T S 2007 Hamiltonian Reduction by Stages (New York: Springer) · Zbl 1129.37001 |
| [12] | Pressley A and Segal G 1988 Loop Groups (Oxford: Clarendon) · Zbl 0638.22009 |
| [13] | Tiglay F and Vizman C 2011 Euler-Poincaré equations on Lie groups and homogeneous spaces, their orbit invariants and applications Lett. Math. Phys.97 45-60 · Zbl 1219.35198 · doi:10.1007/s11005-011-0464-2 |
| [14] | Vladimirov V A, Proctor M R E and Hughes D W 2015 Vortex dynamics of oscillating flows Arnold Math. J.239 113-26 · Zbl 1325.76052 · doi:10.1007/s40598-015-0010-x |
| [15] | Yang C 2017 Multiscale method, central extensions and a generalized Craik-Leibovich equation J. Geom. Phys.116 228-43 · Zbl 1415.34081 · doi:10.1016/j.geomphys.2017.02.004 |
| [16] | Yang C 2016 On the Hamiltonian and geometric structure of the Craik-Leibovich equation (arXiv:1612.00296) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
