Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-16T13:58:51.987Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Hamiltonian structure of evolution equations

Published online by Cambridge University Press:  24 October 2008

Peter J. Olver
Peter J. Olver
Affiliation:
University of Oxford
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikiî on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 71 - 88
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Andersen, I. M. and Duchamp, T. On the existence of global variational principles. University of Utah. (Preprint.)Google Scholar
(2)Benjamin, T. B., Bona, J. L. and Mahoney, J. J.Model equations for long waves in non-linear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972), 47–78.Google Scholar
(3)Cartan, E.Leçons sur les Invariants Intégraux (Paris, A. Hermann et fils, 1922).Google Scholar
(4)Cartan, H. and Eilenberg, S.Homological Algebra (Princeton University Press, 1956).Google Scholar
(5)Gardner, C. S.Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12 (1971), 1548–1551.CrossRefGoogle Scholar
(6)Gel'fand, I. M. and Dikiî, L. A.Asymptotic behaviour of the resolvent of Sturm-Liou-ville equations and the algebra of the Korteweg-de Vries equation. Russ. Math. Surveys 30 (1975), 77–113.CrossRefGoogle Scholar
(7)Gel'fand, I. M. and Dikiî, L. A.A Lie algebra structure in a formal variational calculation. Funk. Anal. 10, 1 (1976), 18–25.Google Scholar
(8)Gel'fand, I. M. and Dikiî, L. A.Fractional powers of operators and Hamiltonian systems. Funk. Anal. 10, 4 (1976), 13–29.Google Scholar
(9)Gel'fand, I. M. and Dikiî, L. A.Resolvents and Hamiltonian systems. Funk. Anal. 11, 2 (1977), 11–27.Google Scholar
(10)Kruskal, M. D., Miura, R. M., Gardner, C. S. and Zabusky, N. J.Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys. 11 (1970), 952–960.CrossRefGoogle Scholar
(11)Kuperschmidt, B. A. Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism. M.I.T. (Preprint.)Google Scholar
(12)Manin, Yu. I.Algebraic aspects of nonlinear differential equations. J. Soviet Math. 11 (1979), 1–122.CrossRefGoogle Scholar
(13)Miura, R. M., Gardner, C. S. and Kruskal, M. D.Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9 (1968), 1204–1209.CrossRefGoogle Scholar
(14)Noether, E.Invariante Variations-Probleme, Kgl. Oer. Wiss. Nachr. Göltingen. Math. Phys. Kl. (1918), 235–257.Google Scholar
(15)Olver, P. J.Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18 (1977), 1212–1215.CrossRefGoogle Scholar
(16)Olver, P. J.Euler operators and conservation laws of the BBM equation. Math. Proc. Cambridge Phil. Soc. 85 (1979), 143–160.CrossRefGoogle Scholar
(17)Olver, P. J. Applications of Lie Groups to Differential Equations, Oxford University Lecture Notes Series (1980).Google Scholar
(18)Olver, P. J. and Shakiban, C.A resolution of the Euler Operator. I. Proc. Amer. Math. Soc. 66 (1978), 223–229.CrossRefGoogle Scholar
(19)Ovsjannikov, L. V.Group Properties of Differential Equations. Translation Bluman, G. W.. (Unpublished.)Google Scholar
(20)Poincaré, H., Les Méthodes Nouvelles de la Mécanique Celeste (Paris, Gauthier-Villars, 1892-1899).Google Scholar
(21)Shadwick, W. F. The Hamilton-Cartan formalism for higher order conserved currents. I. Regular first order Lagrangians. (Preprint.)Google Scholar
(22)Shakiban, C. Ph.D. thesis, Brown University, Providence, R.I. 1979.Google Scholar
(23)Sternberg, S.Lectures on Differential Geometry. (Englewood Cliffs, N.J., Prentice-Hall, 1964).Google Scholar
(24)Sternberg, S.Some preliminary remarks on the formal variational calculus of Gel'fand-Dikiî. Springer Lecture Notes in Math. 676 (1978), 399–407.CrossRefGoogle Scholar
(25)Takens, F. A global version of the inverse problem of the calculus of variations. University of Groningen. (Preprint.)Google Scholar
(26)Tsujishita, T. On variational bicomplexes associated to differential equations. Princeton University. (Preprint.)Google Scholar
(27)Tulczyjew, W. M.The Lagrange complex. Bull. Soc. Math. France 105 (1977), 419–431.Google Scholar
(28)Vainberg, M. M.Variational Methods for the Study of Nonlinear Operators (London, Holden-Day, 1964).Google Scholar
(29)Vinogradov, A. M.On the algebro-geometric foundations of Lagrangian field theory. Soviet Math. Dokl. 18 (1977), 1200–1204.Google Scholar
(30)Vinogradov, A. M.A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints. Soviet Math. Dokl. 19 (1978), 144–148.Google Scholar
(31)Vinogradov, A. M.Hamilton structures in field theory. Soviet Math. Dokl. 19 (1978), 790–794.Google Scholar
(32)Gel'fand, I. M. and Dorfman, I.Ya. Hamiltonian operators and algebraic structures related to them. Funk. Anal. 13 (1979), 13–30. (Russian.)Google Scholar