Tulczyjew triples: From statics to field theory
Katarzyna Grabowska Janusz Grabowski
Journal of Geometric Mechanics 2013, 5(4): 445-472 doi: 10.3934/jgm.2013.5.445
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.
keywords: classical field theory variational calculus. Lagrange formalism Hamiltonian formalism Tulczyjew triple
The supergeometry of Loday algebroids
Janusz Grabowski David Khudaverdyan Norbert Poncin
Journal of Geometric Mechanics 2013, 5(2): 185-213 doi: 10.3934/jgm.2013.5.185
A new concept of Loday algebroid (and its pure algebraic version -- Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.
keywords: Algebroid pseudoalgebra Loday algebra homological vector field Cartan calculus. supercommutative manifold Courant bracket
Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
Janusz Grabowski Katarzyna Grabowska Paweł Urbański
Journal of Geometric Mechanics 2014, 6(4): 503-526 doi: 10.3934/jgm.2014.6.503
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.
keywords: minimal surfaces. Tulczyjew triples Lagrange formalism Hamiltonian formalism double vector bundles variational calculus

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