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Canonical multi-symplectic structure on the total exterior algebra bundle

Bridges, TJ (2006) Canonical multi-symplectic structure on the total exterior algebra bundle Proceedings Royal Society London A, 462 (2069). pp. 1531-1551.

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The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n-dimensional orientable manifold All there is a canonical quadratic form Theta associated with the total exterior algebra bundle on M. On the fibre, which has dimension 2(n), the form Theta can be locally decomposed into n classical symplectic structures. When concatenated, these n-symplectic structures define a partial differential operator, J(partial derivative), which turns out to be a Dirac operator with multi-symplectic structure. The operator J(partial derivative) generalizes the product operator J(d/dt) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The, structure generated by 19 provides a natural setting for analysing a class of covariant nonlinear gradient, elliptic operators. The operator J(partial derivative) is elliptic, and the generalization of Hamiltonian systems, J(partial derivative)Z=del S(Z), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem-find S(Z) for a given elliptic PDE-is shown to be related to a variant of the Legendre transform on k-forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Date : 8 May 2006
DOI : 10.1098/rspa.2005.1629
Uncontrolled Keywords : symplecticity, differential forms, nonlinear elliptic PDEs, Dirac operators, ELLIPTIC SYSTEM, SOLITARY WAVES, EQUATIONS, GEOMETRY, FRONTS
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 17 May 2017 11:22
Last Modified : 10 Jun 2019 13:57

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