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.
The static of smooth maps from the two-dimensional disc to a smooth manifold can be regarded as a simplified
version of the Classical Field Theory. In this paper we construct the Tulczyjew triple for the problem and
describe the Lagrangian and Hamiltonian formalism. We outline also natural generalizations of this approach to
A geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple, is presented. We show the evolution of these concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and ending with Tulczyjew triples for classical field theories illustrated with a few important examples.
Tulczyjew triple for physical systems with configuration manifold equipped with a Lie group structure is constructed and discussed. Systems invariant with respect to group and subgroup actions are considered together with appropriate reductions of the Tulczyjew triple. The theory is applied to free and constrained rigid-body dynamics.
The Lagrangian description of mechanical systems and the Legendre Transformation
(considered as a passage from the Lagrangian to the Hamiltonian
formulation of the dynamics) for point-like objects, for which the
infinitesimal configuration space is $T M$, is based on the existence of
canonical symplectic isomorphisms of double vector bundles $T^* TM$, $T^*T^* M$,
and $TT^* M$, where the symplectic structure on $TT^* M$ is the tangent lift of the canonical symplectic structure $T^* M$.
We show that there exists an analogous picture in the dynamics of objects for which the configuration space is $\wedge^n T M$, if we make use of certain structures of graded bundles of degree $n$, i.e. objects generalizing vector bundles (for which $n=1$). For instance, the role of $TT^*M$ is played in our approach by the manifold $\wedge^nT M\wedge^nT^*M$, which is canonically a graded bundle of degree $n$ over $\wedge^nT M$. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.
We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds.