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Multisymplectic structures and the variational bicomplex

Published online by Cambridge University Press:  04 August 2009

THOMAS J. BRIDGES
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU 2 7XH. e-mail: [email protected], [email protected]
PETER E. HYDON
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU 2 7XH. e-mail: [email protected], [email protected]
JEFFREY K. LAWSON
Affiliation:
Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723, U.S.A.

Abstract

Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Arnold, V. I. and Khesin, B. A. Topological methods in hydrodynamics. Appl. Math. Sciences 125 (Springer-Verlag, 1998).Google Scholar
[2]Abraham, R. and Marsden, J. E.Foundations of Mechanics, Second Edition (Addison-Wesley, 1978).Google Scholar
[3]Anderson, I. M.The Variational Bicomplex, book manuscript. (Utah State University, 1989). http://www.math.usu.edu/~fg_mp/Publications/VB/vb.pdfGoogle Scholar
[4]Anderson, I. M.Introduction to the variational bicomplex. In Mathematical aspects of classical field theory. Contemp. Math. 132 (1992), 5173.CrossRefGoogle Scholar
[5]Binz, A., Śniatycki, J., and Fischer, H. Geometry of classical fields. North-Holland Mathematics Studies 154 (North-Holland, 1988).Google Scholar
[6]Bridges, T. J.Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc. 121 (1997), 147190.CrossRefGoogle Scholar
[7]Bridges, T. J.Toral-equivariant partial differential equations and quasiperiodic patterns. Nonlinearity 11 (1998), 467500.CrossRefGoogle Scholar
[8]Bridges, T. J.Canonical multi-symplectic structure on the total exterior algebra bundle. Proc. Royal Soc. London A 462 (2006), 15311551.Google Scholar
[9]Bridges, T. J. and Laine-Pearson, F. E.Multi-symplectic relative equilibria, multi-phase wavetrains and coupled NLS equations. Stud. Appl. Math. 107 (2001), 137155.CrossRefGoogle Scholar
[10]Cantrijn, F., Ibort, A. and De León, M.On the geometry of multisymplectic manifolds. J. Austral. Math. Soc. (Ser. A) 66 (1999), 303330.Google Scholar
[11]Cullen, M. J. P., Douglas, R. J., Roulstone, I., and Sewell, M. J.Generalized semi-geostrophic theory on a sphere. J. Fluid Mech. 531 (2005), 123157.CrossRefGoogle Scholar
[12]Frankel, T.The Geometry of Physics (Cambridge University Press, 1997).Google Scholar
[13]Gotay, M. J., Isenberg, J., Marsden, J. E., and Montgomery, R. Momentum maps and classical fields. Part I: Covariant field theory, arXiv preprint physics/9801019 (1998).Google Scholar
[14]Hydon, P. E.Multisymplectic conservation laws for differential and differential-difference equations. Proc. Royal Soc. London A 461 (2005), 16271637.Google Scholar
[15]Kanatchikov, I.Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys. 41 (1998), 4990.CrossRefGoogle Scholar
[16]Lawson, J. K.A frame-bundle generalization of multisymplectic geometry. Rep. Math. Phys. 45 (2000), 183205.CrossRefGoogle Scholar
[17]Lawson, J. K.A frame-bundle generalization of multisymplectic momentum mappings. Rep. Math. Phys. 53 (2004), 1937.Google Scholar
[18]de León, M., McLean, M., Norris, L. K., Rey–Roca, A. and Salgado, M. Geometric structures in field theory. Preprint arXiv.org math-ph/0208036 (2002).Google Scholar
[19]Marsden, J. E. and Ratiu, T. S. Introduction to mechanics and symmetry. Texts Appl. Math. 17, Second edition (Springer-Verlag, 1999).Google Scholar
[20]Marshall, J. S.Inviscid Incompressible Flow (John Wiley and Sons, 2001).Google Scholar
[21]Norris, L. K.Generalized symplectic geometry on the frame bundle of a manifold. Proc. Symp. Pure Math. 54 (1993), 435465.CrossRefGoogle Scholar
[22]Paufler, C. and Römer, H.Geometry of Hamiltonian n-vector fields in multisymplectic field theory. J. Geom. Phys. 44 (2002), 5269.CrossRefGoogle Scholar
[23]Saunders, D. J.The Geometry of Jet Bundles. LMS Lecture Note Series 142 (Cambridge University Press, 1989).CrossRefGoogle Scholar
[24]Tulczyjew, W. M. The Euler-Lagrange resolution. In Lecture Notes in Mathematics 836, 2248 (Springer-Verlag, 1980).Google Scholar
[25]Vinogradov, A. M.A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints. Sov. Math. Dokl. 19 (1978), 144148.Google Scholar
[26]Vinogradov, A. M.The C-spectral sequence, Lagrangian formalism and conservation laws I, II. J. Math. Anal. Appl. 100 (1984), 1129.CrossRefGoogle Scholar