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Quasilinear Hamiltonian systems

Published online by Cambridge University Press:  14 November 2011

Jan S. Rogulski
Jan S. Rogulski
Affiliation:
Institute of Mathematics, Warsaw Technical University, Pl. Jedności Robotniczej 1, 00-661 Warszawa, Poland
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Synopsis

We consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 3-4 , 1987 , pp. 195 - 204
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Chernoff, P. R. and Marsden, J. E.. Properties of Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics 425 (Berlin: Springer, 1974).Google Scholar
2Ebin, D. and Marsden, J. E.. Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92 (1970), 102–163.CrossRefGoogle Scholar
3Gawedzki, K.. On the geometrization of the canonical formalism in the classical field theory. Rep. Math. Phys. 3 (1972), 307–326.CrossRefGoogle Scholar
4Haack, W. and Wendland, W.. Lectures on Partial and Pfaffian Differential Equations (Oxford: Pergamon Press, 1971).Google Scholar
5Hughes, T. J. R. and Marsden, J. E.. Mathematical Foundations of Elasticity (New Jersey: Prentice Hall, 1983).Google Scholar
6Kato, T.. Quasilinear Equations of Evolution, with Application to Partial Differential Equations, Lecture Notes in Mathematics 448, pp. 25–70 (Berlin: Springer, 1975).Google Scholar
7Kijowski, J.. A finite dimensional canonical formalism in the classical field theory. Comm. Math. Phys. 30 (1973), 99–128.CrossRefGoogle Scholar
8Kondracki, W.. Geometrization of the classical field theory and its application in Yang-Mills field theory. Rep. on Math. Phys. 16 (1979), 9–47.CrossRefGoogle Scholar
9Palais, R. S.. Foundations of Global Nonlinear Analysis (New York: Benjamin, 1968).Google Scholar