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Multisymplectic structures and the variational bicomplex

Published online by Cambridge University Press:  04 August 2009

THOMAS J. BRIDGES ,
PETER E. HYDON  and
JEFFREY K. LAWSON
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.
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 159 - 178
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), 51–73.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), 147–190.CrossRefGoogle Scholar
[7]Bridges, T. J.Toral-equivariant partial differential equations and quasiperiodic patterns. Nonlinearity 11 (1998), 467–500.CrossRefGoogle Scholar
[8]Bridges, T. J.Canonical multi-symplectic structure on the total exterior algebra bundle. Proc. Royal Soc. London A 462 (2006), 1531–1551.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), 137–155.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), 303–330.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), 123–157.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), 1627–1637.Google Scholar
[15]Kanatchikov, I.Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys. 41 (1998), 49–90.CrossRefGoogle Scholar
[16]Lawson, J. K.A frame-bundle generalization of multisymplectic geometry. Rep. Math. Phys. 45 (2000), 183–205.CrossRefGoogle Scholar
[17]Lawson, J. K.A frame-bundle generalization of multisymplectic momentum mappings. Rep. Math. Phys. 53 (2004), 19–37.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), 435–465.CrossRefGoogle Scholar
[22]Paufler, C. and Römer, H.Geometry of Hamiltonian n-vector fields in multisymplectic field theory. J. Geom. Phys. 44 (2002), 52–69.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, 22–48 (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), 144–148.Google Scholar
[26]Vinogradov, A. M.The C-spectral sequence, Lagrangian formalism and conservation laws I, II. J. Math. Anal. Appl. 100 (1984), 1–129.CrossRefGoogle Scholar