The Tulczyjew triple in field theories of order higher than one
By Katarzyna Grabowska (U. Warsaw)
The geometrical structure known as Tulczyjew triple was used with success in analytical mechanics and first order field theory to describe wide range of systems including systems with constraints or with singular Lagrangians. Starting from basic concepts of variational calculus we derive Tulczyjew triples for the theory of arbitrary given order. The first triple appears as a result of considering a theory of order k + 1 as constrained first order theory with configurations being jets of order k. Constructing the second triple we use reduction to get rid of some nonphysical degrees of freedom. This picture of higher order field theory is covariant and complete: it contains both Lagrangian and Hamiltonian formalisms as well as Euler-Lagrange equations. Geometry of jet bundles is affine rather than linear therefore we have to use the notion of affine duality and affine phase space. The theory is ilustrated by physical examples.