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INTRODUCTION I to MARSDEN and SYMMETRY Alan Weinstein University - PowerPoint PPT Presentation

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INTRODUCTION I to MARSDEN and SYMMETRY Alan Weinstein University of California, Berkeley ICIAM Vancouver July 20, 2011 1 [ slide #1] Jerrold (Jerry) Marsden August 17 1942 September 21, 2010 Berkeley, 1997; George Bergman Websites:

  1. INTRODUCTION I to MARSDEN and SYMMETRY Alan Weinstein University of California, Berkeley ICIAM Vancouver July 20, 2011 1 [ slide #1]

  2. Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 Berkeley, 1997; George Bergman Websites: http://www.cds.caltech.edu/~marsden/remembrances/ (created by W. McKay) obituaries, papers, lectures http://library.caltech.edu/coda/marsden.php publications, with links 2 [ slide #2 ]

  3. 3 [ slide #3 ]

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  5. continued... 5 [ slide #4 ]

  6. Marsden, J. E. (1965) A Theorem on Harmonic Homologies Canad.Math. Bull. 8 (1) , 275-277. ISSN 0008-4395 6 [ slide #5 ]

  7. continued... 7 [ slide #6 ]

  8. Marsden, J. E. (1967) A Correspondence Principle for Momentum Operators Canad.Math. Bull. 10 , 247 – 250 8 [ slide #7 ]

  9. ( continued ) Marsden, J. E. (1967) A Correspondence Principle for Momentum Operators Canad.Math. Bull. 10 , 247 – 250 9 [ slide #8 ]

  10. Marsden, J. E. (1968) Generalized Hamiltonian Mechanics Archive for Rational Mechanics and Analysis, 28 (5) 323 – 361 Our purpose is to generalize Hamiltonian mechanics to the case in which the energy function (Hamiltonian), H , is a distribution (generalized function) in the sense of Schwartz. We follow the same general program as in the smooth case. Familiarity with the smooth case is helpful, although we have striven to make the exposition self-contained, starting from calculus on manifolds. 10 [ slide #9 ]

  11. Marsden, J. E. (1968) Hamiltonian One Parameter Groups Archive for Rational Mechanics and Analysis, 28 (5) , pp. 362 – 396 Our purpose is to give an exposition of the foundations of non-linear conservative mechanical systems with an infinite number of degrees of freedom. Systems we have in mind are the vibrating string, the electromagnetic field and quantum mechanics. These are all linear. We also outline a non-linear example, the coupled Maxwell and Dirac fields. Perfect fluids will be discussed elsewhere. 11 [ slide #9 ]

  12. Marsden, J. E. (1968) Generalized Hamiltonian Mechanics Archive for Rational Mechanics and Analysis, 28 (5) 323 – 361 12 [ slide #10 ]

  13. ( continued ) Marsden, J. E. (1968) Generalized Hamiltonian Mechanics Archive for Rational Mechanics and Analysis, 28 (5) 323 – 361 13 [ slide #10 ]

  14. Marsden, J. E. (1968) Hamiltonian One Parameter Groups Archive for Rational Mechanics and Analysis, 28 (5) 323 – 361 14 [ slide #10 ]

  15. ( continued ) Marsden, J. E. (1968) Hamiltonian One Parameter Groups Archive for Rational Mechanics and Analysis, 28 (5) 323 – 361 15 [ slide #10 ]

  16. Marsden, J. E. (1969) Hamiltonian Systems with Spin Canadian Mathematical Bulletin, pp. 03 – 208 16 [ slide #10 ]

  17. Ebin, David G and Marsden, Jerrold (1970) Groups of diffeomorphisms and the motion of an incompressible fluid Annals of Mathematics, 92 (1) , 102 – 163 17 [ slide #11 ]

  18. Fischer, A. E. and Marsden, Jerrold E. (1972) The Einstein Equations of Evolution—A Geometric Approach Journal of Mathematical Physics, 13 (4) pp. 546 – 568 18 [ slide #11 ]

  19. Marsden, Jerrold E. and Ebin, David G and Fischer, Arthur E. (1972) Diffeomorphisms, hydrodynamics and relativity in the Proceedings of the 13th Biennial Semina of the Canadian Mathematical Congress, pp. 135 – 279 19 [ slide #11 ]

  20. Marsden, J. E. (1973) On Completeness of Homogeneous Pseudo-Riemannian Manifolds Indiana University Mathematics Journal, 22 (11) , pp. 1065 – 1066 M. Guediri and J. Lafontaine (1995) Sur la complétude des variétés pseudo-riemanniennes J. Geom. Phys., 15 (2) , pp. 150 – 158 We discuss completeness for pseudo-riemannian manifolds, and give new examples of non-complete compact manifolds. The former are simply connected, the latter locally homogeneous. 20 [ slide #12 ]

  21. Marsden, Jerrold E. and Weinstein, Alan D. (1974) Reduction of symplectic manifolds with symmetry Reports on Mathematical Physics, 55 (1) , pp. 121 – 130 21 [ slide #13 ]

  22. continued... [ slide #14 ] 22

  23. Arms, Judith M. and Marsden, Jerrold E. and Moncrief, Vincent (1981) Symmetry and Bifucations of Momentum Mappings Communications in Mathematical Physics, 78 (4) , pp. 455 – 478 23 [ slide #15 ]

  24. Marsden, Jerrold E. and Weinstein, Alan D. (1982) The Hamiltonian structure of the Maxwell–Vlasov equations Physcia D, 4 (3) , pp. 394 – 406 Morrison [25] has observed that the Maxwell-Vlasov and Poisson-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. We derive another Poisson structure for these equations by using general methods of symplectic geometry. The main ingredients in our construction are the symplectic structure on the co-adjoint orbits for the group of canonical transformations, and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory. Our Poisson bracket satisfies the Jacobi identity, whereas Morrison's does not [37]. Our construction also shows where canonical variables can be found and can be applied to the Yang-Mills-Vlasov equations and to electromagnetic fluid 24 [ slide #16 ]

  25. Chillingworth, D.R., Marsden, Jerrold E., and Wan, Y. H. (1982) Symmetry and bifucation in three-dimensional elasticity, Part I Archive for Rational Mechanics and Analysis, 80 (4) , pp. 295 – 331 25 [ slide #17 ]

  26. Krishnaprasad, P. S. and Marsden, Jerrold E. (1987) Hamiltonian structures and stability for rigid bodies with flexible attachments Archive for Rational Mechanics and Analysis, 98 (1) , pp. 71 – 93 Posberg, T.A., Krishnaprasad, Perinkulam S. and Marsden, Jerrold E. (1986) Stability Analysis of a Rigid Body with a Flexible Attachment Using the Energy–Casimir Method In Differential Geometry: the interface between pure and applied mathematics American Mathematical Society, Providence, RI San Antonio, TX (1987) , Contemporary Mathematics 68 , pp. 71 – 93 26 [ slide #18 ]

  27. ( continued ) Posberg, T.A., Krishnaprasad, Perinkulam S. and Marsden, Jerrold E. (1986) Stability Analysis of a Rigid Body with a Flexible Attachment Using the Energy–Casimir Method In Differential Geometry: the interface between pure and applied mathematics American Mathematical Society, Providence, RI San Antonio, TX (1987) , Contemporary Mathematics 68 , pp. 71 – 93 27 [ slide #18 ]

  28. ( continued ) Posberg, T.A., Krishnaprasad, Perinkulam S. and Marsden, Jerrold E. (1986) Stability Analysis of a Rigid Body with a Flexible Attachment Using the Energy–Casimir Method In Differential Geometry: the interface between pure and applied mathematics American Mathematical Society, Providence, RI San Antonio, TX (1987) , Contemporary Mathematics 68 , pp. 71 – 93 28 [ slide #18 ]

  29. Marsden, J. E. , O’Reilly, O. M., Wicklin, F. J. and Zombro, B.W. (1991) Symmetry, Stability, Geometric Phases, and Mechanical Integators Nonlinear Science Today, 1 (1) , pp. 4 – 11, 1 (2) , pp. 14 – 21 29 [ slide #19 ]

  30. continued... 30 [ slide #19 ]

  31. continued... 31 [ slide #19 ]

  32. Gotay, Mark J., Marsden, Jerrold E., and Montgomery, Richard (2004) Momentum Maps and Classical Fields Part I: Covariant Field Theory arXiv physics/9801019v2 [math-ph] ( Submitted on 16 Jan 1998 (v1), last revised 19 Aug 2004 (this version, v2) This is the first paper of a five part work in which we study the Lagrangian and Hamiltonian structure of classical field theories with constraints. Our goal is to explore some of the connections between initial value constraints and gauge transformations in such theories (either relativistic or not). To do this, in the course of these four papers, we develop and use a number of tools from symplectic and multisymplectic geometry. Of central importance in our analysis is the notion of the ``energy- momentum map'' associated to the gauge group of a given classical field theory. We hope to demonstrate that many different and apparently unrelated facets of field theories can be thereby tied together and understood in an essentially new way. In Part I we develop some of the basic theory of classical fields from a spacetime covariant viewpoint. We begin with a study of the covariant Lagrangian and Hamiltonian formalisms, on jet bundles and multisymplectic manifolds, respectively. Then we discuss symmetries, conservation laws, and Noether's theorem in terms of ``covariant momentum maps.'' Comments: LaTeX2e, 68 pages, 1 figure, GIMMsy 1; Updated, with minor revisions and corrections Subjects: Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th) Cite as: arXiv:physics/9801019v2 [math-ph] 32 [ slide #20 ]

  33. continued... 33 [ slide #20 ]

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