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MP: Theoretische und Mathematische Grundlagen der Physik

MP 6: Classical Field Theory

MP 6.1: Vortrag

Mittwoch, 28. März 2001, 17:15–17:45, HS VIII

Geometry of n-vectors in multisymplectic field theory — •Cornelius Paufler and Hartmann Römer — Albert-Ludwigs-Universität Freiburg im Breisgau, Fakultät für Physik, Hermann-Herder-Straße 3, 79104 Freiburg im Breisgau

The framework of multisymplectic geometry allows to formulate classical field theories on spaces of finite dimension. The central idea is that in field theory, solutions of the field equations are sections over space-time, while in classical mechanics solutions of the equations of motion can be viewed as sections over the time axis. Hence, one can generalise Hamiltonian mechanics such that space and time directions are treated on equal footing, i.e. serve as evolutionary parameters.

In this work, we investigate the geometrical interpretation of n-vector fields (n-fold antisymmetric tensor products of vector fields) which are to replace the usual Hamiltonian vector fields of classical mechanics. It turns out that those multi-vectors span the tangent space at every point of a given solution and, conversely, n-vectors of certain Hamiltonians can be integrated to yield foliations of solutions of the Euler-Lagrange equations. Finally, the results allow yet another justification of the generalised Schrödinger equation for multisymplectic geometry which has been found recently by I. Kanatchikov.