Tulczyjew triples in higher derivative field theory

Katarzyna Grabowska; Luca Vitagliano

doi:10.3934/jgm.2015.7.1

Article Contents

2015, Volume 7, Issue 1: 1-33. Doi: 10.3934/jgm.2015.7.1

This issue Previous Article Next Article Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations

Tulczyjew triples in higher derivative field theory

Katarzyna Grabowska^1, and
Luca Vitagliano^2,

1.
Department of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa
2.
Department of Mathematics, University of Salerno, and Istituto Nazionale di Fisica Nucleare, GC Salerno, Via Giovanni Paolo II, 123, 84084 Fisciano (SA)

Received: August 31, 2014

Revised: November 30, 2014

Published: February 2015

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Abstract

The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems, including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus, we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrarily high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.

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Mathematics Subject Classification: Primary: 70S05, 70H03, 70H05; Secondary: 58A20, 53D05.

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