introduction - mathtube.org€¦ · alan weinstein 1 introduction i to marsden and symmetry [ slide...
TRANSCRIPT
![Page 1: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/1.jpg)
University of California, Berkeley
ICIAM Vancouver July 20, 2011
Alan Weinstein
1
IINTRODUCTIONto MARSDEN
and SYMMETRY
[ slide #1]
![Page 2: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/2.jpg)
2
Jerrold (Jerry) MarsdenAugust 17 1942 – September 21, 2010
Websites:http://www.cds.caltech.edu/~marsden/remembrances/
(created by W. McKay)obituaries, papers, lectures
http://library.caltech.edu/coda/marsden.phppublications, with links
Berkeley, 1997; George Bergman
[ slide #2 ]
![Page 3: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/3.jpg)
3[ slide #3 ]
![Page 4: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/4.jpg)
4[ slide #4 ]
![Page 5: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/5.jpg)
continued...
5[ slide #4 ]
![Page 6: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/6.jpg)
Marsden, J. E. (1965) A Theorem on Harmonic Homologies
Canad.Math. Bull. 8 (1), 275-277. ISSN 0008-4395
[ slide #5 ] 6
![Page 7: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/7.jpg)
continued...
7[ slide #6 ]
![Page 8: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/8.jpg)
8
Marsden, J. E. (1967)A Correspondence Principle for Momentum Operators
Canad.Math. Bull. 10, 247–250
[ slide #7 ]
![Page 9: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/9.jpg)
( continued )
9[ slide #8 ]
Marsden, J. E. (1967)A Correspondence Principle for Momentum Operators
Canad.Math. Bull. 10, 247–250
![Page 10: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/10.jpg)
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.
[ slide #9 ]
![Page 11: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/11.jpg)
11
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.
Marsden, J. E. (1968)Hamiltonian One Parameter Groups
Archive for Rational Mechanics and Analysis, 28 (5), pp. 362–396
[ slide #9 ]
![Page 12: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/12.jpg)
12
Marsden, J. E. (1968)Generalized Hamiltonian Mechanics
Archive for Rational Mechanics and Analysis, 28 (5) 323–361
[ slide #10 ]
![Page 13: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/13.jpg)
( continued )
13[ slide #10 ]
Marsden, J. E. (1968)Generalized Hamiltonian Mechanics
Archive for Rational Mechanics and Analysis, 28 (5) 323–361
![Page 14: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/14.jpg)
14[ slide #10 ]
Marsden, J. E. (1968)Hamiltonian One Parameter Groups
Archive for Rational Mechanics and Analysis, 28 (5) 323–361
![Page 15: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/15.jpg)
( continued )
15
Marsden, J. E. (1968)Hamiltonian One Parameter Groups
Archive for Rational Mechanics and Analysis, 28 (5) 323–361
[ slide #10 ]
![Page 16: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/16.jpg)
16
Marsden, J. E. (1969)Hamiltonian Systems with Spin
Canadian Mathematical Bulletin, pp. 03–208
[ slide #10 ]
![Page 17: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/17.jpg)
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
[ slide #11 ]
![Page 18: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/18.jpg)
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
[ slide #11 ]
![Page 19: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/19.jpg)
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 theCanadian Mathematical Congress, pp. 135–279
[ slide #11 ]
![Page 20: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/20.jpg)
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.
[ slide #12 ]
![Page 21: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/21.jpg)
21
Marsden, Jerrold E. and Weinstein, Alan D. (1974)Reduction of symplectic manifolds with symmetry
Reports on Mathematical Physics, 55 (1), pp. 121–130
[ slide #13 ]
![Page 22: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/22.jpg)
continued...
22[ slide #14 ]
![Page 23: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/23.jpg)
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
[ slide #15 ]
![Page 24: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/24.jpg)
24
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 generalmethods 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
Marsden, Jerrold E. and Weinstein, Alan D. (1982)The Hamiltonian structure of the Maxwell–Vlasov equations
Physcia D, 4 (3), pp. 394–406
[ slide #16 ]
![Page 25: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/25.jpg)
25[ slide #17 ]
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
![Page 26: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/26.jpg)
26[ slide #18 ]
Krishnaprasad, P. S. and Marsden, Jerrold E. (1987)Hamiltonian structures and stability for rigid bodies
with flexible attachmentsArchive 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
![Page 27: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/27.jpg)
( continued )
27[ slide #18 ]
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
![Page 28: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/28.jpg)
( continued )
28[ slide #18 ]
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
![Page 29: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/29.jpg)
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
[ slide #19 ]
![Page 30: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/30.jpg)
continued...
30[ slide #19 ]
![Page 31: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/31.jpg)
continued...
31[ slide #19 ]
![Page 32: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/32.jpg)
32[ slide #20 ]
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]
Gotay, Mark J., Marsden, Jerrold E., and Montgomery, Richard (2004)Momentum Maps and Classical Fields
Part I: Covariant Field TheoryarXiv physics/9801019v2 [math-ph]
(Submitted on 16 Jan 1998 (v1), last revised 19 Aug 2004 (this version, v2)
![Page 33: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/33.jpg)
continued...
33[ slide #20 ]
![Page 34: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/34.jpg)
34
Kobilarov, Marin, Marsden, Jerrold E., and Sukhatme, Gaurav S. (2010)Geometric Discretization of Nonholonomic Symmetries
Discrete and Continuous Dynamical Systems Series S, 3 (1), pp. 61–84, ISBN 1937–1632
ABSTRACT. The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach. We focus on systems with symmetries, controllable shape (internal variables), and nonholonomic constraints. Motivated by the abundance of important models from science and engineering with such properties, we propose numerical methods specifically designed to account for their special geometric structure. At the core of the formulation lies a discrete variational principle that respects the structure of the state space and provides a framework for constructing accurate and numerically stable integrators. The dynamics of the systems we study is derived by vertical and horizontal splitting of the varia- tional principle with respect to a nonholonomic connection that encodes the kinematic constraints and symmetries. We formulate a discrete analog of this principle by evaluating the Lagrangian and the connection at selected points along a discretized trajectory and derive discrete momentum equation and discrete reduced Euler-Lagrange equations resulting from the splitting of this principle. A family of nonholonomic integrators that are general, yet simple and easy to implement, are then obtained and applied to two examples-the steered robotic car and the snakeboard. Their numerical advantages are confirmed through comparisons with standard methods.[ slide #21]
![Page 35: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/35.jpg)
( continued )
35[ slide #21]
Kobilarov, Marin, Marsden, Jerrold E., and Sukhatme, Gaurav S. (2010)Geometric Discretization of Nonholonomic Symmetries
Discrete and Continuous Dynamical Systems Series S, 3 (1), pp. 61–84, ISBN 1937–1632
![Page 36: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/36.jpg)
( continued )
36[ slide #21]
Kobilarov, Marin, Marsden, Jerrold E., and Sukhatme, Gaurav S. (2010)Geometric Discretization of Nonholonomic Symmetries
Discrete and Continuous Dynamical Systems Series S, 3 (1), pp. 61–84, ISBN 1937–1632
![Page 37: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/37.jpg)
( continued )
37[ slide #21]
Kobilarov, Marin, Marsden, Jerrold E., and Sukhatme, Gaurav S. (2010)Geometric Discretization of Nonholonomic Symmetries
Discrete and Continuous Dynamical Systems Series S, 3 (1), pp. 61–84, ISBN 1937–1632
![Page 38: INTRODUCTION - mathtube.org€¦ · Alan Weinstein 1 INTRODUCTION I to MARSDEN and SYMMETRY [ slide #1] 2 Jerrold (Jerry) Marsden August 17 1942 – September 21, 2010 ... starting](https://reader036.vdocuments.net/reader036/viewer/2022062611/6130ec4a1ecc515869446815/html5/thumbnails/38.jpg)
continued...
38[ slide #22]