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In this paper we study coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of coisotropic reduction is motivated by the fact that these dynamics can always be interpreted as Lagrangian or Legendrian submanifolds. Furthermore, Lagrangian or Legendrian submanifolds can be reduced by a coisotropic one. [ABSTRACT FROM AUTHOR]

By means of the Jacobi structure associated with a contact structure, we use the so-called evolution vector field to propose a new characterization of isolated thermodynamical systems with friction, a simple but important class of thermodynamical systems which naturally satisfy the first and second laws of thermodynamics, i.e. total energy preservation of isolated systems and non-decreasing total entropy, respectively. We completely clarify its qualitative dynamics, the underlying geometrical structures and we also show how to apply discrete gradient methods to numerically integrate the evolution equations for these systems. [ABSTRACT FROM AUTHOR]

Walter Noll's trailblazing constitutive theory of material defects in smoothly uniform bodies is recast in the language of Lie groupoids and their associated Lie algebroids. From this vantage point the theory is extended to non-uniform bodies by introducing the notion of singular material distributions and the physically cognate idea of graded uniformity and homogeneity. [ABSTRACT FROM AUTHOR]

We present several results on the inverse problem and equivalent contact Lagrangian systems. These problems naturally lead to consider smooth transformations on the z variable (i.e., reparametrizations of the action). We present the extended contact Lagrangian systems to formalize this notion. With this structure we define horizontal equivalence of Lagrangians, which generalizes the symplectic case. We also present some results on the inverse problem for extended contact systems. [ABSTRACT FROM AUTHOR]

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations. [ABSTRACT FROM AUTHOR]