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Citations Previous Up Next From References: 47 From Reviews: 3 Article MR719663 (84k:58081) 58F05 (58G40 70E15 73C99 76N99 76W05 78A99) Marsden, Jerrold E. (1-CA); Raţiu, Tudor (1-AZ); Weinstein, Alan (1-CA) Semidirect products and reduction in mechanics. Trans. Amer. Math. Soc. 281 (1984), no. 1, 147–177. Let G be a Lie group with the Lie algebra g, and let ρ be a left representation of G on a vector space V . Then, one has a semidirect product S = G ×ρ V given by the operation (g1 , v1 )(g2 , v2 ) = (g1 g2 , v1 + ρ(g1 )v2 ). The authors construct a Lie-Poisson structure on the dual space g ∗ which makes g ∗ isomorphic with the reduced Poisson manifold T ∗ G/G. This is used for a detailed study of reductions in the case of a semidirect product, which, in turn, is applied to the following theories: the heavy top, compressible fluids, magnetohydrodynamics, elasticity, electromagnetic coupling, and multifluid plasmas. A dual Lie algebra version of these theories is thereby obtained. Reviewed by I. Vaisman References 1. Abraham and J. E. Marsden [1978], Foundations of mechanics, 2nd ed., Addison-Wesley, Reading, Mass. MR0515141 (81e:58025) 2. P. Bretherton [1970], A note on Hamilton’s principle for perfect fluids, J. Fluid Mech. 44, 19-31. 3. E. Dzyaloshinskii and G. E. Volovick [1980], Poisson brackets in condensed matter physics, Ann. Physics 125, 67-97. MR0565078 (81a:81129) 4. G. Ebin and J. Marsden [1970], Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92, 102-163. MR0271984 (42 #6865) 5. Guillemin and S. Sternberg [1980], The moment map and collective motion, Ann. Physics 127, 220-253. MR0576424 (81g:58011) 6. Guillemin and S. Sternberg [1980] [1982] Symplectic techniques in physics (book in preparation). 7. Holm and B. Kupershmidt [1983], Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas and elasticity, Physica D (to appear). cf. MR 85e:58045 8. J. Holmes and J. E. Marsden [1983], Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana Univ. Math. J. 32, 273-310. MR0690190 (84h:58056) 9. Hughes, T. Kato and J. Marsden [1977], Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 273-294. 10. Kazhdan, B. Kostant and S. Sternberg [1978], Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31, 481-508. MR0478225 (57 #17711) 11. Kupershmidt [1982], Discrete Lax equations and differential-difference equations (lecture notes). 12. Lie [1890], Theorie der Transformationsgruppen, Zweiter Abschnitt, Teubner, Leipzig. 13. E. Marsden [1976], Well-posedness of the equations of a non-homogeneous perfect fluid, Comm. Partial Differential Equations 1, 215-230. MR0405493 (53 #9286) 14. E. Marsden [1976] [1981], Lectures on geometric methods in mathematical physics, CBMSNSF Regional Conf. Ser. in Appl. Math., No. 37, SIAM, Philadelphia, Pa. MR0619693 (82j:58046) 15. E. Marsden and T. J. R. Hughes [1983], Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, N.J. 16. E. Marsden and A. Weinstein [1974], Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121-130. MR0402819 (53 #6633) 17. E. Marsden and A. Weinstein [1974] [1982a], The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D 4, 394-406. MR0657741 (84b:82037) 18. E. Marsden and A. Weinstein [1974] [1982a] [1982b], Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D (to appear). 19. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R. G. Spencer [1983], Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-ISIMM Sympos. (Modern Developments in Analytical Mechanics, Torino, June 7-11, 1982) (to appear). cf. MR 86f:58062 20. J. Morrison [1982], Poisson brackets for fluids and plasmas, Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems (M. Tabor and Y. M. Treve, eds.), AIP Conf. Proc. (La Jolla Institute, 1981) 88, 13-46. MR0694249 (84c:76002) 21. J. Morrison and J. M. Greene [1980], Noncanonical hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics, Phys. Rev. Letters, 45, 790-794. MR0581650 (81g:76114) 22. Ratiu [1980], Euler-Poisson equations on Lie algebras, Thesis, Univ. of Calif., Berkeley. 23. Ratiu [1980] [1981], Euler-Poisson equations on Lie algebras and the N -dimensional heavy rigid body, Proc. Nat. Acad. Sci. U.S.A. 78, 1327-1328. MR0607652 (82g:58032) 24. Ratiu [1980] [1981] [1982], Euler-Poisson equations on Lie algebras and the N -dimensional heavy rigid body, Amer. J. Math. 104, 409-448. MR0654413 (84c:58045a) 25. H. Rawnsley [1975], Representations of a semi-direct product by quantization, Math. Proc. Cambridge. Philos. Soc. 75, 345-350. MR0387499 (52 #8341) 26. L. Seliger and G. B. Whitham [1968], Variational principles in continuum mechanics, Proc. Roy. Soc. London Ser. A 305, 1-25. 27. G. Spencer [1982], The Hamiltonian structure of multi-species fluid electrodynamics, Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems (M. Tabor and V. M. Treve, eds.), AIP Conf. Proc. (La Jolla Institute, 1981) 88, 121-126. MR0694249 (84c:76002) 28. Sternberg [1977], On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci. 74, 5253-5254. MR0458486 (56 #16686) 29. M. Vinogradov and B. Kupershmidt [1977], The structure of Hamiltonian mechanics, Russian Math. Surveys 32, 177-243. MR0501143 (58 #18573) 30. Weinstein [1978], A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2, 417-420. MR0507025 (80e:58023) 31. Weinstein [1978] [1982], The local structure of Poisson manifolds, J. Differential Geometry (to appear). cf. MR 86i:58059 Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 1984, 2013