Canonical multisymplectic structure on the total exterior algebra bundle
Bridges, TJ (2006) Canonical multisymplectic structure on the total exterior algebra bundle Proceedings Royal Society London A, 462 (2069). pp. 15311551.
Full text not available from this repository.Abstract
The aim of this paper is to construct multisymplectic 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 ndimensional 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 nsymplectic structures define a partial differential operator, J(partial derivative), which turns out to be a Dirac operator with multisymplectic 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 problemfind S(Z) for a given elliptic PDEis shown to be related to a variant of the Legendre transform on kforms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with nontrivial 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  
URI:  http://epubs.surrey.ac.uk/id/eprint/831106 
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