Download Marsden, Jerrold E. (1-CA)

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