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Multisymplectic structures and the variational bicomplex

  • THOMAS J. BRIDGES (a1), PETER E. HYDON (a1) and JEFFREY K. LAWSON (a2)
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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[4]I. M. Anderson Introduction to the variational bicomplex. In Mathematical aspects of classical field theory. Contemp. Math. 132 (1992), 5173.

[6]T. J. Bridges Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc. 121 (1997), 147190.

[7]T. J. Bridges Toral-equivariant partial differential equations and quasiperiodic patterns. Nonlinearity 11 (1998), 467500.

[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.

[15]I. Kanatchikov Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys. 41 (1998), 4990.

[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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