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 arbitrary 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.
Highlights
1.1 Variational calculus in StaticsFrom a mathematical point of view, calculus of variations is a theory providing tools for finding extremals, or stationary points, of functionals, i.e. maps from a set of functions to real numbers
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
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
Summary
From a mathematical point of view, calculus of variations is a theory providing tools for finding extremals, or stationary points, of functionals, i.e. maps from a set of functions to real numbers. Given two systems with the same configuration manifold and constitutive sets C1 and C2 we can answer the question whether or not q is an equilibrium point for the composite system. This happens precisely when Cq1 ∩ Cq2 = ∅. The above ideas apply efficiently to other theories as mechanics or field theory as well To see this we shall specify a configuration space Q, a set of processes (or at least infinitesimal processes), the set of functions on Q (to define regular systems), the set of covectors T∗Q (to define constitutive sets). The main aim of the present paper is showing how things work in the case of higher derivative field theory
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