  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.
