This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J. L. Lagrange, “Mémoire sur la théorie des variations de éléments des planètes,” Mém. Cl. Sci. Math. Phys. Inst. France, 1–72 (1808).
A. Weinstein, “Symplectic Geometry,” Bull. Am. Math. Soc. 5 1–13 (1981).
H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Vols. 1, 2, 3 (Gauthier-Villars, Paris, 1892; Dover, New York, 1957).
E. Noether, Nachr. Gesell. Wiss. Göttingen 2, 136 (1918).
P. A. M. Dirac, The Principles of Quantum Mechanics, (Oxford University Press, 1958).
As in L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vols. 1–10 (Pergamon, New York, 1960-1981).
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition (Benjamin/Cummings, Reading, Mass., 1978).
V. I. Arnol'd, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978).
J. R. Cary, “Lie transform perturbation theory of Hamiltonian systems,” Phys. Rep. 79, 131 (1981).
P. R. Chernoff and J. E. Marsden, Properties of Infinite Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 425 (Springer, New York, 1974).
P. J. Morrison and J. M. Greene, “Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics,” Phys. Rev. Lett. 45, 790–794 (1980).
J. E. Marsden and A. Weinstein, “Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,” Physica 7D, 305–323 (1983).
J. E. Marsden, T. Ratiu, and A. Weinstein, “Semi-direct products and reduction in mechanics,” Trans. Am. Math. Soc. 281, pp. 147–177 (1984).
P. J. Morrison, “The Maxwell-Vlasov equations as a continuous Hamiltonian systems,” Phys. Lett. 80A, 383–396 (1980).
J. Marsden and A. Weinstein, “The Hamiltonian structure of the Maxwell-Vlasov equations,” Physica 4D, 394–406 (1982).
W. Pauli, Die Allgemeinen Prinzipien der Wellenmeckanik, Hand. der Phys., Vol 24, Part 1 (Springer, Berlin, 1933); General Principles of Quantum Mechanics (Springer, Berlin, 1981).
D. Holm and B. Kupershmidt, “Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity,” Physica D.
J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity (Prentice-Hall, Englewood Cliffs, New Jersey, 1983).
R. G. Spencer and A. N. Kaufman, “Hamiltonian structure of two-fluid plasma dynamics,” Phys. Rev. A25, 2437–2439 (1982).
J. Gibbons, D. D. Holm, and B. Kupershmidt, “Gauge-invariant Poisson brackets for chromohydrodynamics,” Phys. Lett. 90A, 281–283 (1982).
D. D. Holm and B. A. Kupershmidt, “Poisson structures of superfluids,” submitted to Phys. Lett. A.
L. Faddeev and V. E. Zakharov, “Korteweg-de Vries as a completely integrable Hamiltonian system,” Functional Anal. Appl. 5, 280 (1971).
D. D. Holm, J. E. Marsden, T. Ratiu, and A. Weinstein, “A priori estimates for nonlinear stability of fluids and plasmas,” Preprint, 1984.
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, (Cambridge University Press, Cambridge, 1984).
S. Omohundro, “Hamiltonian structures in perturbation theory,” submitted to J. of Math. Phys. (1984).
S. Omohundro, “A Hamiltonian approach to wave modulation,” Poster presented at Sherwood Plasma Theory Meeting, Lake Tahoe, April 1984 (in preparation for publication).
A. H. Nayfeh, Perturbation Methods (John Wiley, New York, 1973).
J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics (Springer, New York, 1981).
A. Weinstein, “The local structure of Poisson manifolds,” J. Diff. Geom. 18, 523–557 (1983).
J. E. Marsden, Lectures on Geometric Methods in Mathematical Physics, CBMS-NSF Reg. Conf. Series 37 (PA:SIAM, Philadelphia, 1981).
P. Olver, “Hamiltonian perturbation theory and water waves,” Fluids and Plasmas: Geometry and Dynamics, J. Marsden, Ed., Contemporary Mathematics, 28, 231 (1984).
J. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” Rep. Math. Phys. 5, 121–130 (1974).
V. Arnol'd, “Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications a hydrodynamique des fluids parfaits,” Ann. Inst. Fourier Grenoble 16(1) 319–361 (1966).
J. Kijowski and W. Tulczyjew, A symplectic framework for field theories, Lecture Notes in Physics 107 (Springer, New York, 1979).
V. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, 1983).
M. Kruskal, “Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic,” J. Math. Phys. 3, 806–828 (1962).
R. Kubo, H. Ichimura, T. Usui, and N. Hashitsume, Statistical Mechanics (North-Holland, Amsterdam, 1965).
G. Whitham, Linear and Nonlinear Waves (John Wiley, New York, 1974).
R. Littlejohn, J. Plasma Physics 29, 111 (1983).
N. van Kampen, “Constraints,” Preprint, March 1983.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Omohundro, S. (1984). Geometric Hamiltonian structures and perturbation theory. In: Sáenz, A.W., Zachary, W.W., Cawley, R. (eds) Local and Global Methods of Nonlinear Dynamics. Lecture Notes in Physics, vol 252. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0018331
Download citation
DOI: https://doi.org/10.1007/BFb0018331
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16485-2
Online ISBN: 978-3-540-39824-0
eBook Packages: Springer Book Archive