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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 148, Issue 1
  • January 2010, pp. 159-178

Multisymplectic structures and the variational bicomplex

  • THOMAS J. BRIDGES (a1), PETER E. HYDON (a1) and JEFFREY K. LAWSON (a2)
  • DOI: http://dx.doi.org/10.1017/S0305004109990259
  • Published online: 04 August 2009
Abstract
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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[4]I. M. Anderson Introduction to the variational bicomplex. In Mathematical aspects of classical field theory. Contemp. Math. 132 (1992), 5173.

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[8]T. J. Bridges Canonical multi-symplectic structure on the total exterior algebra bundle. Proc. Royal Soc. London A 462 (2006), 15311551.

[9]T. J. Bridges and F. E. Laine-Pearson Multi-symplectic relative equilibria, multi-phase wavetrains and coupled NLS equations. Stud. Appl. Math. 107 (2001), 137155.

[14]P. E. Hydon Multisymplectic conservation laws for differential and differential-difference equations. Proc. Royal Soc. London A 461 (2005), 16271637.

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[16]J. K. Lawson A frame-bundle generalization of multisymplectic geometry. Rep. Math. Phys. 45 (2000), 183205.

[17]J. K. Lawson A frame-bundle generalization of multisymplectic momentum mappings. Rep. Math. Phys. 53 (2004), 1937.

[19]J. E. Marsden and T. S. Ratiu Introduction to mechanics and symmetry. Texts Appl. Math. 17, Second edition (Springer-Verlag, 1999).

[22]C. Paufler and H. Römer Geometry of Hamiltonian n-vector fields in multisymplectic field theory. J. Geom. Phys. 44 (2002), 5269.

[24]W. M. Tulczyjew The Euler-Lagrange resolution. In Lecture Notes in Mathematics 836, 2248 (Springer-Verlag, 1980).

[26]A. M. Vinogradov The C-spectral sequence, Lagrangian formalism and conservation laws I, II. J. Math. Anal. Appl. 100 (1984), 1129.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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