Geometric mechanics, Part I: Dynamics and symmetry

Geometric Mechanics, Part I: Dynamics and Symmetry

Volumn , Issue , 2008, Pages 1-354

  Holm, Darryl D ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

CALCULATIONS; COMPUTATION THEORY; EDUCATION; GEOMETRY; LIE GROUPS; LINEAR ALGEBRA; MECHANICS; METAL DRAWING; TEACHING;

COMPUTATIONAL MANIPULATION; GEOMETRIC MECHANICS; HAMILTONIAN FORMULATIONS; LAGRANGE VARIATIONAL PRINCIPLE; LIE GROUP SYMMETRIES; PHYSICAL APPLICATION; REDUCTION BY SYMMETRIES; RESONANT OSCILLATORS;

STUDENTS;

EID: 84967566823     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/P557     Document Type: Book

References (184)

1. [AbHoMaRa1986] Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. and Ratiu, T. S. [1986] Nonlinear stability analysis of stratified ideal fluid equilibria. Phil Trans. Roy. Soc. (London) A 318, 349-409.
2. [AbMa1978] Abraham, R. and Marsden, J. E. [1978], Foundations of Mechanics. Addison-Wesley, second edition.
3. [AcHoKoTi1997] Aceves, A., Holm, D. D., Kovacic, G. and Timofeyev, I. [1997] Homoclinic orbits and chaos in a secondharmonic generating optical cavity. Phys. Lett. A 233, 203-208.
4. [AbMaRa1988] Abraham, R., Marsden, J. E. and Ratiu, T. S. [1988], Manifolds, Tensor Analysis, and Applications. Volume 75 of Applied Mathematical Sciences. Springer-Verlag, second edition.
5. [AlLuMaRo1998] Alber, M. S., Luther, G. G., Marsden, J. E. and Robbins, J. M. [1998] Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction. Phys. D, 123, pp. 271-290.
6. [AlLuMaRo1999] Alber, M. S., Luther, G. G., Marsden, J. E. and Robbins, J. M. [1999] Geometry and control of three-wave interactions, in The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun. 24, AMS, Providence, RI, pp. 55-80.
7. [AlBaNe1990] Allen, J. S., Barth, J. A. and Newberger, P. A. [1990] On intermediate models for barotropic continental shelf and slope flow fields. Part I: Formulation and comparison of exact solutions. J. of Phys. Oceanog. 20, 10171042
8. [AlHo1996] Allen, J. S., and Holm, D. D. [1996] Extendedgeostrophic Hamiltonian models for rotating shallow water motion. Physica D, 98 229-248.
9. [AlHoNe2002] Allen, J. S., Holm, D. D., and Newberger, P. A. [2002] Toward an extended-geostrophic Euler-Poincaré model for mesoscale oceanographic flow. In Large-Scale Atmosphere-Ocean Dynamics 1: Analytical Methods and Numerical Models. Edited by J. Norbury & I. Roulstone, Cambridge University Press: Cambridge, pp. 101-125.
10. [AlEb1975] Allen, L., and Eberly, J. H. [1975] Optical Resonance and Two-Level Atoms, Wiley: NewYork. (Reprinted by Dover, Mineola, NY, 1987)
11. [AmCZ96] Ambrosetti, A. and Coti Zelati, V. [1996], Periodic Solutions of Singular Lagrangian Systems. Birkhäuser
12. [AnMc1978] Andrews, D. G. and McIntyre, M. E. [1978] An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609-646.
13. [Ar350BC] Aristotle [350BC] Physics.
14. [Ar1966] Arnold, V. I. [1966], Sur la géometrie differentialle des groupes de Lie de dimiension infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier, Grenoble 16, 319-361.
15. [Ar1979] Arnold, V. I. [1979], Mathematical Methods of Classical Mechanics. Volume 60 of Graduate Texts in Mathematics. Springer-Verlag.
16. [Ar1994] Arnold, V. I. [1994] Topological Invariants of Plane Curves and Caustics. Amer. Math. Soc. University Lecture Series 5.
17. [ArKh1992] Arnold, V. I. and Khesin, B. A. [1992] Topological methods in hydrodynamics. Ann. Rev. of Fluid Mech. 24, 145-166.
18. [ArKh1998] Arnold, V. I and Khesin, B. A. [1998] Topological Methods in Hydrodynamics. Springer: New York.
19. [Be1986] Benci, V. [1986], Periodic solutions of Lagrangian systems on a compact manifold, Journal of Differential Equations 63, 135-161.
20. [BeBuHo1994] Berman, G. P., Bulgakov, E. N. and Holm, D. D. [1994] Crossover-Time in Quantum Boson and Spin Systems, Springer Lecture Notes in Physics: Monographs m21, Springer: New York.
21. [BlBr92] Blanchard, P. and Bruning, E. [1992], Variational Methods in Mathematical Physics. Springer Verlag.
22. [Bl2004] Bloch, A. M. (with the collaboration of: Baillieul, J., Crouch, P. E. and Marsden, J. E.) [2004] Nonholonomic Mechanics and Control, Springer-Verlag NY.
23. [BlBrCr1997] Bloch, A. M., Brockett, R. W. and Crouch, P. E. [1997] Double bracket equations and geodesic flows on symmetric spaces. Comm. Math Phys 187, 357-373.
24. [BlCr1997] Bloch, A. M. and Crouch, P. E. [1996] Optimal control and geodesic flows. Systems and Control Letters 28, 65-72.
25. [BlCrHoMa2001] Bloch, A.M., Crouch, P. E., Holm, D. D. and Marsden, J. E. [2001] Proc. of the 39th IEEEE Conference on Decision and Control, Proc. CDC 39, 1273-1279.
26. [BlCrMaRa1998] Bloch, A. M., Crouch, P. E., Marsden, J. E. and Ratiu, T. S. [1998] Discrete rigid body dynamics and optimal control. Proc. CDC IEEE, 37, 2249-2254.
27. [Bl1965] Bloembergen, N. [1965] Nonlinear optics, Benjamin. New York.
28. [BoWo1965] Born, M. and Wolf, E. [1965] Principles of optics (Oxford: Pergamon Press) p. 31.
29. [Bo71] Bourbaki, N. [1971] Variétés différentielles et analytiques. Fascicule de résultats. Hermann.
30. [Bo1989] Bourbaki, N. [1989] Lie Groups and Lie Algebras. Chapters 1-3. Springer Verlag.
31. [Br1964] Bretherton, F. P. [1964] Resonant interactions between waves: The case of discrete oscil lations, J. Fluid Mech., 20, 457-479.
32. [Br1895] Bruns, H. [1895] Leipzig. Math. Phys. Ber. 21, 325.
33. [Bu 1993] Buchdahl, H. A. [1993] An Introduction to Hamiltonian Optics. New York: Dover.
34. [Ca1995] Calogero, F. [1995] An integrable hamiltonian system. Phys. Lett. A 201, 306-310.
35. [CaFr1996] Calogero, F. and Francoise, J.-P. [1996] An integrable hamiltonian system. J. Math. Phys. 37, 2863-2871.
36. [CaHo1993] Camassa, R. and Holm, D. D. [1993] An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661-64.
37. [CeMaPeRa2003] Cendra, H., Marsden, J. E., Pekarsky, S., and Ratiu, T. S. [2003] Variational principles for Lie-Poisson and Hamilton-Poincaré equations, Moskow Math. Journ. 3(3), 833-867.
38. [ChMa1974] Chernoff, P. R. and Marsden, J. E. [1974] Properties of Infinite Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 425, Springer-Verlag.
39. [CuBa1997] Cushman, R. H. and Bates, L. M. [1997] Global Aspects of Integrable Systems. Birkhäuser.
40. 2 Molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy. Phys. Rev. Lett. 93, 024302.
41. [DaBaBi1992] Dandoloff, R., Balakrishnan, R. and Bishop, A. R. [1992] Two-level systems: space curve formalism, Berry’s phase and Gauss-Bonnet theorem. J. Phys. A: Math. Gen. 25, L1105-L1110.
42. [DaHo1992] David, D. and Holm, D. D. [1992] Multiple Lie-Poisson structures, reductions, and geometric phases for the Maxwell-Bloch traveling-wave equations. J. Nonlin. Sci. 2, 241-262.
43. [DaHoTr1989] David, D., Holm, D. D. and Tratnik, M. V. [1989] Horseshoe chaos in a periodically perturbed polarized optical beam. Phys. Lett. A 138, 29-36.
44. [DaHoTr1990] David, D., Holm, D. D. and Tratnik, M. V. [1990] Hamiltonian chaos in nonlinear polarization dynamics. Phys. Rep. 187, 281-367.
45. [DeMe1993] Dellnitz, M. and Melbourne, I. [1993] The equivariant Darboux theorem. Lectures in Appl. Math. 29, 163-169.
46. [Dr2007] Dragt, A. J. [2007] Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics. In preparation.
47. [DrJo1989] Drazin, P. G. and Johnson, R. S. [1989] Solitons: An Introduction. Cambridge University Press.
48. [DuNoFo1995] Dubrovin, B., Novikov, S. P. and Fomenko, A. T. [1995] Modern Geometry I, II, III, Volumes 93, 104, 124 of Graduate Texts in Mathematics. Springer-Verlag.
49. [DuGiCu2004] Dullin, H., Giacobbe, A. and Cushman, R. H. [2004] Monodromy in the resonant swing spring. Physica D 190, 15.
50. [DuGoHo2001] Dullin, H., Gottwald, G. and Holm, D. D. [2001] An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 194501-04.
51. [DuGoHo2003] Dullin, H., Gottwald, G. and Holm, D. D. [2003] Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73-95.
52. [DuGoHo2004] Dullin, H., Gottwald, G. and Holm, D. D. [2004] On asymptotically equivalent shallow water wave equations. Physica D 190, 1-14.
53. [EbMa1970] Ebin, D. G. and Marsden, J. E. [1970] Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92, 102-163.
54. [El2006] Ellis, D. C. P. [2006] The Hopf fibration and the 1 − 1 resonant optical travelling wave pulse. MSci Project, Imperial College London.
55. [Er1942] Ertel, H. [1942] Ein neuer hydrodynamischer Wirbelsatz. Met. Z., Braunschweig, pp. 277-281.
56. [FeVeHe1957] Feynman, R., Vernon, F. and Hellwarth, R. [1957] Appl. Phys. 28, 49-52.
57. [Fl1963] Flanders, H. [1963] Differential Forms with Applications to the Physical Sciences, Academic Press, New York.
58. [FoHo1991] Fordy, A. P. and Holm, D. D. [1991] A tri-Hamiltonian formulation of the self-induced transparency equations. Phys. Lett A 160, 143-148.
59. [FrHo2001] Fringer, O. B. and Holm, D. D. [2001] Integrable vs nonintegrable geodesic soliton behavior. Physica D 150, 237-263.
60. [Ga1632] Galilei, G. [1632] Dialogue Concerning the Two Chief World Systems: The Ptolemaic and the Copernican. http://en.wikipedia.org/wiki/Dialogue Concerning the Two Chief World Systems
61. [Ge1961] Gelfand, I. M. [1961] Lectures on Linear Algebra. Interscience, New York.
62. [GeDo1979] Gelfand, I. M. and Dorfman, I. Ya. R. [1979] Funct. Anal. Appl. 13, 248.
63. [GiHoKu1982] Gibbons, J., Holm, D. D. and Kupershmidt, B. A. [1982], Gauge-invariant Poisson brackets for chromohydrodynamics, Phys. Lett. A 90, 281-283.
64. [GjHo1996] Gjaja, I. and Holm, D. D. [1996] Self-consistent wavemean flow interaction dynamics and its Hamiltonian formulation for a rotating stratified incompressible fluid. Physica D 98, 343-378.
65. [GuHo1983] Guckenheimer, J. and Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin.
66. [GuSt1984] Guillemin, V. and Sternberg, S. [1984] Symplectic Techniques in Physics. Cambridge University Press.
67. [HaMe1998] Haller, G. and Mezic, I. [1998] Reduction of threedimensional, volume-preserving flows with symmetry. Nonlinearity 11, 319-339.
68. [Ha1830] Hamilton, W. R. [1830] Supplement to an essay on the theory of systems of rays. Transactions of the Royal Irish Academy, 16, part 1 pp. 1-61.
69. [Ha1837] Hamilton, W. R. [1837] Third supplement to an essay on the theory of systems of rays. Transactions of the Royal Irish Academy, 17, part 1, pp. 1-144.
70. [Ha2006] Hanson, A. J. [2006] Visualizing Quaternions. MKElsevier.
71. [HaMi1977] Hasegawa, A. and Mima, K. [1997] Pseudo-threedimensional turbulence in magnetized nonuniform plasmas. Phys. Fluids 21, 87-92.
72. [He1858] Helmholtz, H., [1858] On integrals of the hydrodynamical equations which express vortex motion. Transl. Tait, P. G. [1867] Phil. Mag. 33, no. 4, 485-512.
73. [Ho2002a] Holm, D. D. [2002] Euler-Poincaré dynamics of perfect complex fluids. In Geometry, Mechanics, and Dynamics: in honor of the 60th birthday of Jerrold E. Marsden edited by P. Newton, P. Holmes and A. Weinstein. Springer, pp. 113-167.
74. [Ho2002b] Holm, D. D. [2002] Variational principles for Lagrangian averaged fluid dynamics. J. Phys. A: Math. Gen. 35, 1-10.
75. [Ho2005] Holm, D. D. [2005] The Euler-Poincaré variational framework for modeling fluid dynamics, in Geometric Mechanics and Symmetry: The Peyresq Lectures, edited by J. Montaldi and T. Ratiu, London Mathematical Society Lecture Notes Series 306, Cambridge University Press (2005).
76. [HoKo1991] Holm, D. D. and Kovacic, G. [1991] Homoclinic chaos for ray optics in a fiber. Physica D 51, 177-188.
77. [HoKo1992] Holm, D. D. and Kovacic, G. [1992] Homoclinic chaos in a laser-matter system. Physica D 56, 270-300.
78. [HoKoSu1991] Holm, D. D., Kovacic, G. and Sundaram, B. [1991] Chaotic laser-matter interaction. Phys. Lett. A 154, 346-352.
79. [HoKoWe1995] Holm, D. D., Kovacic, G. and Wettergren, T. A. [1995] Near integrability and chaos in a resonant-cavity laser model, Phys. Lett. A 200, 299-307.
80. [HoKoWe1995] Holm, D. D., Kovacic, G. and Wettergren, T. A. [1996] Homoclinic orbits in the Maxwell-Bloch equations with a probe, Phys. Rev. E 54, 243-256.
81. [HoLy2002] Holm, D. D. and Lynch, P. [2002], Stepwise precession of the resonant swinging spring. SIAM J. Applied Dyn. Syst. 1, (1) 44-64.
82. [HoMa1991] Holm, D. D. and Marsden, J. E. [1991] The rotor and the pendulum. in Symplectic Geometry and Mathematical Physics, P. Donato, C. Duval, J. Elhadad, G. M. Tuynman, ed., Prog. in Math. Vol. 99, Birkhauser: Boston, pp. 189-203.
83. [HoMa2004] Holm, D. D. and Marsden, J. E. [2004] Momentum maps and measure valued solutions (peakons, filaments, and sheets) of the Euler-Poincare equations for the diffeomorphism group. In The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein, pp. 203-235, Progr. Math., 232, J.E. Marsden and T.S. Ratiu, Editors, Birkhäuser Boston, Boston, MA. http://arxiv.org/abs/nlin.CD/0312048
84. [HoMaRa1998a] Holm, D. D., Marsden, J. E. and Ratiu, T. S. [1998a] The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. in Math. 137, 1-81.
85. [HoMaRa1998b] Holm, D. D. Marsden, J. E. and Ratiu, T. S. [1998b] Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80, 4173-4177.
86. [HoMaRa2002] Holm, D. D. Marsden, J. E. and Ratiu, T. S. [2002] The Euler-Poincaré equations in geophysical fluid dynamics. In Large-Scale Atmosphere-Ocean Dynamics 2: Geometric Methods and Models. Edited by J. Norbury and I. Roulstone, Cambridge University Press: Cambridge, pp. 251-299.
87. [HoRaTrYo2004] Holm, D. D., Rananather, J. T., Trouvé, A. and Younes, L. [2004] Soliton dynamics in computational anatomy. NeuroImage 23, S170-178 (2004). http://arxiv.org/abs/nlin.SI/0411014
88. [HoMaRaWe1985] Holm, D. D., Marsden, J. E., Ratiu, T. S. and Weinstein, A. [1985] Nonlinear stability of fluid and plasma equilibria. Physics Reports 123 1-116.
89. [HoSt2002] Holm, D. D. and Staley, M. F. [2002] Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs. http://arxiv.org/abs/nlin.CD/0202059.
90. [HoWo1991] Holm, D. D. and Wolf, K. B. [1991] Lie-Poisson description of Hamiltonian ray optics. Physica D 51, 189-199.
91. [Ho1931] Hopf, H. [1931] Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen 104, 637-665.
92. [HoHa1994] Horton, W. and Hasegawa, A. [1994] Quasi-twodimensional dynamics of plasmas and fluids. Chaos 4, 227-251.
93. [HuZh1994] Hunter, J. K. and Zheng, Y. [1994] On a completely integrable nonlinear hyperbolic variational equation. Physica D 79, 361-386.
94. [Is1999] Isham, C. J. [1999] Differential Geometry for Physicists. World Scientific Lecture Notes in Physics 61.
95. [JoSa98] José, J. V. and Saletan, E. J. [1998] Classical Dynamics: A Contemporary Approach. Cambridge University Press.
96. [Jo1998] Jost, J. [1998] Riemannian Geometry and Geometric Analysis. University Text. Springer-Verlag, second edition.
97. [KaKoSt1978] Kazhdan, D., Kostant, B., and Sternberg, S. [1978] Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31, 481-508.
98. [Ke1869] Kelvin, L., [1869] On vortex motion. Trans. Roy. Soc. Edin. 25, 217-260; Maximum and minimum energy in vortex motion. Mathematical and Physical Papers, vol. 4, Cambridge Univ. Press, 172-183.
99. [KhMis03] Khesin, B. and Misiolek, G. [2003] Euler equations on homogeneous spaces and Virasoro orbits, Adv. in Math. 176, 116-144.
100. [KnHj2001] Knudsen, J. M. and Hjorth, P. G. [2001] Elements of Newtonian Mechanics: Including Nonlinear Dynamics. Springer-Verlag, New York, third edition, second printing.
101. [Ko1984] Koiller, J. [1984] A mechanical system with a “wild” horseshoe. J. Math. Phys. 25, 1599-1604.
102. [Ko1966] Kostant, B. [1966] Orbits, symplectic structures and representation theory. Proc. US-Japan Seminar on Diff. Geom., Kyoto. Nippon Hyronsha, Tokyo 77.
103. [KrSc2001] Kruse, H. P., Schreule, J. and Du, W. [2001] A twodimensional version of the CH equation. In Symmetry and Perturbation Theory: SPT 2001 Edited by D. Bambusi, G. Gaeta and M. Cadoni. World Scientific: New York, pp 120-127.
104. [Ku1999] Kuipers, J. B. [1999] Quaternions and Rotation Sequences: a Primer with Applications to Orbits, Aerospace and Virtual Reality. Princeton University Press.
105. [Ku1976] Kummer, M. [1976] On resonant nonlinearly coupled oscillators with two equal frequencies. Commun. Math. Phys. 48, 53-79 (1976).
106. [Ku1978] Kummer, M. [1978] On resonant classical Hamiltonians with two equal frequencies. Commun. Math. Phys. 58, 85-112.
107. [Ku1981] Kummer, M. [1981] On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana Univ. Math. J. 30, 281-291.
108. [Ku1986] Kummer, M. [1986] in Local and Global Methods in Nonlinear Dynamics, Lecture Notes in Physics Vol. 252, edited by A. V. Sáenz, Springer-Verlag, New York, pp. 19-31.
109. [La1999] Lang, S. [1999] Fundamentals of Differential Geometry. Volume 191 of Graduate Texts in Mathematics. Springer-Verlag, New York.
110. [Le2003] Lee, J. [2003] Introduction to Smooth Manifolds, Springer-Verlag.
111. [LiMa1987] Libermann, P. and Marle, C.-M. [1987] Symplectic Geometry and Analytical Mechanics. Reidel.
112. [Lie1890] Lie, S. [1890] Theorie der Transformationsgruppen. Zweiter Abschnitt. Teubner.
113. [LoGi1967] Longuet-Higgins, M. S. and Gill, A. E. [1967] Resonant interactions between planetary waves. Proc. Roy. Soc. Edinburgh Sect. A. 299, 120-140.
114. [Lo1963] Lorenz, E. N. [1963] Deterministic non-periodic flow. J. Atmospheric Sci. 20, 130-141.
115. [LuAlMaRo2000] Luther, G. G., Alber, M. S., Marsden, J. E. and Robbins, J. M. [2000] Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media, J. Opt. Soc. Amer. B Opt. Phys. 17, 932-941.
116. [Ly2002a] Lynch, P. [2002] Resonant motions of the threedimensional elastic pendulum, Internat. J. Non-Linear Mech. 37, 345-367.
117. [Ly2002b] Lynch, P. [2002] The swinging spring: A simple model for atmospheric balance, in Large-Scale Atmosphere-Ocean Dynamics: Vol II: Geometric Methods and Models, Cambridge University Press, Cambridge, UK, pp. 64-108.
118. [Man1976] Manakov, S. V. [1976] Note on the integration of Euler’s equations of the dynamics of and n-dimensional rigid body. Funct. Anal. and its Appl. 10, 328-329.
119. [Mar1976] Marle, C.-M. [1976] Symplectic manifolds, dynamical groups, and Hamiltonian mechanics, in Differential Geometry and Relativity, Cahen, M. and Flato, M. eds., D. Reidel, Boston, 249-269.
120. [Ma1981] Marsden, J. E. [1981] Lectures on Geometric Methods in Mathematical Physics. Volume 37, SIAM, Philadelphia.
121. [Ma1992] Marsden, J. E. [1992] Lectures on Mechanics. Volume 174 of London Mathematical Society Lecture Note Series. Cambridge University Press.
122. [MaHu1983] Marsden, J. E. and Hughes, T. J. R. [1983] Mathematical Foundations of Elasticity. Prentice Hall. Reprinted by Dover Publications, NY, 1994.
123. [MaMiOrPeRa2007] Marsden, J. E., Misiolek, G., Ortega, J.-P., Perlmutter, M., and Ratiu, T. S. [2007] Hamiltonian Reduction by Stages. Lecture Notes in Mathematics, 1913. Springer-Verlag.
124. [MaMoRa1991] Marsden, J. E., Montgomery, R., and Ratiu, T. S. [1991] Reduction, symmetry, and phases in mechanics. Memoirs Amer. Math. Soc. 88, (436) 1-110.
125. [MaRa1994] Marsden, J. E. and Ratiu, T. S. [1994] Introduction to Mechanics and Symmetry. Volume 75 of Texts in Applied Mathematics, second printing of second edition 2003. Springer-Verlag.
126. [MaRa95] Marsden, J. E. and Ratiu, T. S. [2003] Geometric Fluid Dynamics. Unpublished notes.
127. [MaRa03] Marsden, J. E. and Ratiu, T. S. [2003] Mechanics and Symmetry. Reduction Theory. In preparation.
128. [MaRaWe84a] Marsden, J. E., Ratiu, T. S., and Weinstein, A. [1984a] Semidirect products and reduction in mechanics. Trans. Amer. Math. Soc. 281, (1) 147-177.
129. [MaRaWe84b] Marsden, J. E., Ratiu, T. S., and Weinstein, A. [1984b] Reduction and Hamiltonian structures on duals of semidirect product Lie algebras. Contemporary Math., 28, 55-100.
130. [MaWe74] Marsden, J. E. and Weinstein, A. [1974] Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121-130.
131. [MaWe83] Marsden, J. E. and Weinstein, A. [1983] Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D, 7, 305-323.
132. [MaWil989] Mawhin, J. and Willem, M. [1989] Critical Point Theory and Hamiltonin Systems. Volume 74 of Applied Mathematical Sciences. Springer-Verlag, second edition.
133. [McSa1995] McDuff, D. and Salamon, D. [1995] Introduction to Symplectic Topology. Clarendon Press.
134. [MeDe1993] Melbourne, I. and Dellnitz, M. [1993] Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group. Proc. Camb. Phil. Soc. 114, 235-268.
135. [Mi1963] Milnor, J. [1963] Morse Theory. Princeton University Press.
136. [MiFo1978] Mishchenko, A. S. and Fomenko, A. T. [1978] Euler equations on finite dimensional Lie groups. Izv. Acad. Nauk SSSR, Ser. Matem. 42, (2) 396-415 (Russian); English translation: Math. USSR-Izv. 12, (2) 371-389.
137. [Mi2002] Misiolek, G. [2002] Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal., 12, 1080-1104.
138. [Mo1969] Moffatt, H. K. [1969] The degree of knottedness of tangled vortex lines. J. Fluid. Mech. 106, 117-129.
139. [Mo1981] Moffatt, H. K. [1981] Some developments in the theory of turbulence. J. Fluid Mech. 106, 27-47.
140. [Mo1985] Moffatt, H. K. [1985] Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology, Part 1. J. Fluid Mech. 159, 359-378; Part 2 [1986] J. Fluid Mech. 166, 359-378.
141. [MoTs1992] Moffatt, H. K. and Tsinober, A. [1992] Helicity in laminar and turbulent flow. Ann. Rev. Fluid Mech. 24, 281-312.
142. [Mo1991] Montgomery, R. [1991] How much does the rigid body rotate? A Berry’s phase from the 18th Century, Am. J. Phys. 59, 394-398.
143. [Na1973] Nambu, Y. [1973] Generalized Hamiltonian mechanics, Phys. Rev. D 7, 2405-2412.
144. [No1918] Noether, E. [1918] Nachrichten Gesell. Wissenschaft. Göttingen 2, 235.
145. C. H. Kimberling [1972] Am. Math. Monthly 79, 136.
146. [OcoRo1998] O’Connor, J. J. and Robertson, E. F., [1998] Sir William Rowan Hamilton, http://www-groups.dcs.st-and. ac.uk/_history/Mathematicians/Hamilton.html
147. [Ol2000] Olver, P. J. [2000] Applications of Lie Groups to Differential Equations. Springer: New York.
148. [OrRa2004] Ortega, J.-P. and Ratiu, T. S. [2004] Momentum Maps and Hamiltonian Reduction. Volume 222 of Progress in Mathematics. Birkhäuser.
149. [Ot1993] Ott, E. [1993] Chaos in Dynamical Systems, Cambridge University Press, Cambridge, UK, p. 385.
150. [OKh87] Ovsienko, V. Y. and Khesin, B. A. [1987] Korteweg-de Vries superequations as an Euler equation. Funct. Anal. Appl., 21, 329-331.
151. [Pa1968] Palais, R. [1968] Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam.
152. [Po1892] Poincaré H. [1892] Theorie Mathematique de la Lumiere, (ed.) George Carre, Paris, p. 275.
153. [Po1901] Poincaré, H. [1901] Sur une forme nouvelle des équations de la méchanique C.R. Acad. Sci. 132, 369-371.
154. [Ra1980] Ratiu, T. [1980] The motion of the free n-dimensional rigid body. Indiana U. Math. J. 29, 609-627.
155. [RaTuSbSoTe2005] Ratiu, T. S., Tudoran, R., Sbano, L., Sousa Dias, E., Terra, G. [2005] A crash course in geometric mechanics, in Geometric Mechanics and Symmetry: The Peyresq Lectures, edited by J. Montaldi and T. Ratiu, London Mathematical Society Lecture Notes Series 306, Cambridge University Press.
156. [SaCa1981] Sarlet, W. and Cantrijn, F. [1981] Generalizations of Noether’s theorem in classical mechanics. SIAM Review 23, 467-494.
157. [Sc1987] Schmid, R. [1987] Infinite Dimensional Hamiltonian Systems. Bibliopolis.
158. [Se1992] Serre, J.-P. [1992] Lie Algebras and Lie Groups, Volume 1500 of Lecture Notes in Mathematics. Springer-Verlag.
159. [ShWi1989] Shapere, A. and Wilcek, F. [1989] Geometric Phases in Physics, World Scientific: Singapore.
160. [Sh1984] Shen, Y. R. [1984] The Principles of Nonlinear Optics, Wiley-lnterscience, New York.
161. 1 metric and mean hydrodynamics J. Funct. Anal. 160, 337-365.
162. [Sh2000] Shkoller, S. [2000] Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid J. Diff. Geom. 55, 145-191.
163. [Si1913] Silberstein, L. [1913] Vectorial Mechanics, MacMillan and Co., Limited. London. 2nd ed. (1926).
164. [Sm1970] Smale, S. [1970] Topology and mechanics, Inv. Math. 10, 305-331; 11, 45-64.
165. [So1970] Souriau, J. M. [1970] Structure des Systèmes Dynamiques, Dunod, Paris.
166. [Sp1982] Sparrow, C. [1982] The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer-Verlag, New York, p. 269.
167. [Sp1979] Spivak, M. [1979] Differential Geometry, Volume I. New printing with corrections. Publish or Perish, Inc. Houston, Texas.
168. [St1852] Stokes, G. G. [1852] Trans. Camb. Phil. Soc. 9, 399.
169. [Sy1937] Synge, J. L. [1937] Geometrical Optics: An Introduction to Hamilton’s Method, Cambridge U. Press, London.
170. [Sy1937] Synge, J. L. [1937] Hamilton’s method in geometrical optics, J. Opt. Soc. Am. 27, 75-82. http://www.opticsinfobase.org/abstract.cfm?URI= josa-27-2-75
171. [Sy1937] Synge, J. L. [1960] Classical Dynamics, in Handbuch der Physik, edited by S. Flügge Springer-Verlag, Berlin, Vol. III/1, p. 11.
172. [Ta1994] Takhtajan, L. [1994] On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295-315.
173. [Ur2003] Urbantke, H. K. [2003] The Hopf fibration - seven times in physics. J. Geom. Phys. 46 125-150.
174. [Va1996] Vaisman, I. [1996] Lectures on the Geometry of Poisson Manifolds. Volume 118 of Progress in Mathematics, Birkhäuser.
175. [ViGo1933] Vitt, A. and Gorelik, G. [1933] Kolebaniya uprugogo mayatnika kak primer kolebaniy dvukh parametricheski svyazannykh linejnykh sistem, Zh. Tekh. Fiz. (J. Tech. Phys.), 3, pp. 294307. Available in English translation: Oscillations of an Elastic Pendulum as an Example of the Oscillations of Two Parametrically Coupled Linear Systems, Translated by Lisa Shields with an Introduction by Peter Lynch, Historical Note 3, Met Eireann, Dublin, 1999.
176. [Wa1983] Warner, F. W. [1983] Foundation of Differentiable Manifolds and Lie Groups. Volume 94 of Graduate Texts in Mathematics, Springer-Verlag.
177. [We1983] Weinstein, A. [1983a] Sophus Lie and symplectic geometry, Exposition Math. 1, 95-96.
178. [We1983b] Weinstein, A. [1983b] The local structure of Poisson manifolds. Journ. Diff. Geom 18, 523-557.
179. [We2002] Weinstein, A. [2002] Geometry of momentum (preprint); ArXiv:math/SG0208108 v1.
180. [WeFiOt1980] Wersinger, J.-M., Finn, J. M. and Ott, E. [1980] Bifurcations and strange behavior in instability saturation by nonlinear mode coupling. Phys. Rev. Lett. 44, 453-456.
181. [Wh1974] Whitham, G. B. [1974] Linear and Nonlinear Waves, John Wiley and Sons, New York.
182. [Wi1988] Wiggins, S. [1988] Global Bifurcations and Chaos - Analytical Methods, Springer. Berlin.
183. [Wo2004] Wolf, K. B. [2004] Geometric Optics on Phase Space, Springer. ISBN 3540220399.
184. [Wo1958] Woltjer, L. [1958] A theorem on force-free magnetic fields. Proc. Nat. Acad. Sci. USA 44, 489-491.