Variational integrators and time-dependent lagrangian systems

This paper presents a method to construct variational integrators for time-dependent lagrangian systems. The resulting algorithms are symplectic, preserve the momentum map associated with a Lie group of symmetries and also describe the energy variation.Comment: 8 page

Towards a Hamilton-Jacobi Theory for Nonholonomic Mechanical Systems

In this paper we obtain a Hamilton-Jacobi theory for nonholonomic mechanical systems. The results are applied to a large class of nonholonomic mechanical systems, the so-called \v{C}aplygin systems.Comment: 13 pages, added references, fixed typos, comparison with previous approaches and some explanations added. To appear in J. Phys.

Discrete variational integrators and optimal control theory

A geometric derivation of numerical integrators for optimal control problems is proposed. It is based in the classical technique of generating functions adapted to the special features of optimal control problems.Comment: 17 page

Tulczyjew's triples and lagrangian submanifolds in classical field theories

In this paper the notion of Tulczyjew's triples in classical mechanics is extended to classical field theories, using the so-called multisymplectic formalism, and a convenient notion of lagrangian submanifold in multisymplectic geometry. Accordingly, the dynamical equations are interpreted as the local equations defining these lagrangian submanifolds.Comment: 29 page

Geometric numerical integration of nonholonomic systems and optimal control problems

A geometric derivation of numerical integrators for nonholonomic systems and optimal control problems is obtained. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems and optimal control problems.Comment: 6 pages, 1 figure. Submitted to IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Sevilla 200

A new geometric setting for classical field theories

A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists. The main advantage of this algorithm is that the second order condition is automatically included.Comment: 22 page

Simulating Nonholonomic Dynamics

This paper develops different discretization schemes for nonholonomic mechanical systems through a discrete geometric approach. The proposed methods are designed to account for the special geometric structure of the nonholonomic motion. Two different families of nonholonomic integrators are developed and examined numerically: the geometric nonholonomic integrator (GNI) and the reduced d'Alembert-Pontryagin integrator (RDP). As a result, the paper provides a general tool for engineering applications, i.e. for automatic derivation of numerically accurate and stable dynamics integration schemes applicable to a variety of robotic vehicle models

Constrained Variational Calculus for Higher Order Classical Field Theories

We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of application are studied, in particular, applications to the geometrical description of optimal control theory for partial differential equations.Comment: 25 pages; 4 diagram