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Multisymplectic Reduction for Proper Actions

Published online by Cambridge University Press:  20 November 2018

Jędrzej Śniatycki
Jędrzej Śniatycki*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta email: [email protected]
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Abstract

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We consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied.

Keywords

58J7035A30Dedonder equationmultisymplectic structurereductionsymmetriesvariational problems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 3 , 01 June 2004 , pp. 638 - 654
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Binz, E., Śniatycki, J. and Fischer, H., Geometry of Classical Fields. Elsevier, Amsterdam, 1988.Google Scholar
[2] Bridges, T. J. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A 284(2001), 184–193.Google Scholar
[3] Cartan, E., Les èspaces metriques fondés sur la notion d'aire. Actualitès Sc. et Industr. 72, Herman, Paris, 1933.Google Scholar
[4] Lopez, M. Castrillón, Ratiu, T. S. and Shkoller, S., Reduction in Principal Fiber Bundles: Covariant Euler-Poincaré Equations. Proc. Amer.Math. Soc. 128(2000), 2155–2164.Google Scholar
[5] Cendra, H., Marsden, J. E. and Ratiu, T. S.: Geometric mechanics, Lagrangian reduction and nonholonomic systems. In: Mathematics Unlimited—2001 and Beyond (eds. Engquist, B. and Schmid, W.), Springer, Berlin, 2001, 221–273.Google Scholar
[6] Cushman, R. and Bates, L., Global Aspects of Classical Integrable Systems. Birkhäuser, Basel, 1997.Google Scholar
[7] Cushman, R. and Śniatycki, J., Hamiltonian mechanics on principal bundles. C. R. Math. Rep. Acad. Sci. Canada 21(1999), 60–64.Google Scholar
[8] Cushman, R. and Śniatycki, J., Differential structure of orbit spaces. Canad. J. Math. 53(2001), 715–755.Google Scholar
[9] Dedonder, Th., Théorie invariantive du calcul des variations. Bull. Acad. de Belg., 1929.Google Scholar
[10] Duistermaat, J. J. and Kolk, J. A. C., Lie Groups. Springer Verlag, New York, 1999.Google Scholar
[11] Fernandez, A. P., Garcia, P. L. and Rodrigo, C., Lagrangian reduction and constrained variational calculus. In: Proceedings of the IX FallWorkshop on Geometry and Physics (Vilanova i la Getru, July, 2000), Real Sociedad Matematica Española, 2001, 53–64.Google Scholar
[12] Funk, P., Variationsrechnung und ihre Anwendung in Physik und Technik. Springer Verlag, Berlin, 1962 Google Scholar
[13] Gotay, M. J., Isenberg, J. and Marsden, J. E., Momentum Maps and Classical Relativistic Fields, Part I: Covariant Field Theory. MSRI preprint, 1997.Google Scholar
[14] Gotay, M. J., Isenberg, J. and Marsden, J. E., Momentum Maps and Classical Relativistic Fields, Part II: Canonical Analysis of Field Theories. MSRI preprint, 1999.Google Scholar
[15] Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics. Cambridge University Press, Cambridge, 1984.Google Scholar
[16] Kijowski, J. and Tulczyjew, W. M., A Symplectic Framework for Field Theories. Lecture Notes in Physics 107, Springer Verlag, Berlin, 1979.Google Scholar
[17] Lepage, Th. H. J., Acad. Roy. Belg. Bull. Cl. Sci. V. Sér. 22(1936), 716.Google Scholar
[18] Marsden, J. E. and Scheurle, J., The reduced Euler-Lagrange equations. Fields Inst. Commun. 1(1993), 139–164.Google Scholar
[19] Marsden, J. E., Patrick, G. W. and Shkoller, S., Multisymplectic geometry, variational integrators and nonlinear PDEs. Commun. Math. Phys. 199(1998), 351–395.Google Scholar
[20] Marsden, J. E., Pekarsky, S., Shkoller, S. and West, M., Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38(2001), 253–284.Google Scholar
[21] Śniatycki, J., Regularity of constraints and reduction in the Minkowski space Yang-Mills-Dirac theory. Ann. Inst. H. Poincaré 70(1999), 2–7–293.Google Scholar
[22] Śniatycki, J., Schwarz, G. and Bates, L., Yang-Mills and Dirac fields in a bag, constraints and reduction. Commun. Math. Phys. 176(1996), 95–115.Google Scholar
[23] Śniatycki, J. and Weinstein, A., Reduction and quantization for singular momentum mappings. Lett. Math. Phys. 7(1983), 155–161.Google Scholar