A geometrical analysis of the field equations in field theory

In this review paper we give a geometrical formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and non-uniqueness of solutions, as well as their integrability.Comment: 14 pages. LaTeX file. This is a review paper based on previous works by the same author

Singular Lagrangian Systems on Jet Bundles

The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we give a basis of constraint functions in the Lagrangian and Hamiltonian sides. An additional description of constraints is also given considering at the same time compatibility, stability and second-order condition problems. Finally, a classification of the constraints in first and second class is obtained using a cosymplectic geometry setting. Using the second class constraints, a Dirac bracket is introduced, extending the well-known construction by Dirac.Comment: 65 pages. LaTeX fil

Global controllability tests for geometric hybrid control systems

Hybrid systems are characterized by having an interaction between continuous dynamics and discrete events. The contribution of this paper is to provide hybrid systems with a novel geometric formulation so that controls can be added. Using this framework we describe some new global controllability tests for hybrid control systems exploiting the geometry and the topology of the set of jump points, where the instantaneous change of dynamics take place. Controllability is understood as the existence of a feasible trajectory for the system joining any two given points. As a result we describe examples where none of the continuous control systems are controllable, but the associated hybrid system is controllable because of the characteristics of the jump set.Comment: 27 pages, 5 figure