D. Martín de Diego

Geometric Description of Vakonomic and Nonholonomic Dynamics. Comparison of Solutions

We treat the vakonomic dynamics with general constraints within a new geometric framework, which can be useful in the study of optimal control problems. We compare our formulation with the one of Vershik and Gershkovich in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics. Examples illustrating the theory are treated.

A SURVEY OF LAGRANGIAN MECHANICS AND CONTROL ON LIE ALGEBROIDS AND GROUPOIDS

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

Geometric integrators and nonholonomic mechanics

A geometric derivation of nonholonomic integrators is developed. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems. The theoretical methodology and the integrators obtained are different from those obtained in Cortes and Martinez [“Nonholonomic integrators,” Nonlinearity 14, 1365–1392 (2001)]. In the case of mechanical systems with linear constraints a family of geometric integrators preserving the nonholonomic constraints is given.

Nonholonomic constraints: A new viewpoint

The purpose of this paper is to show that, at least for Lagrangians of mechanical type, nonholonomic Euler-Lagrange equations for a nonholonomic linear constraint D may be viewed as non-constrained Euler-Lagrange equations but on a new (generally not Lie) algebroid structure on D. The proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of Neoether Theorem producing actual first integrals in case of symmetries.

Singular Lagrangian systems and variational constrained mechanics on Lie algebroids

The purpose of this article is to describe Lagrangian mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles …). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.

SYMMETRIES IN CLASSICAL FIELD THEORY

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.

Geometric aspects of nonholonomic field theories

A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.

A unified framework for mechanics: Hamilton–Jacobi equation and applications

In this paper, we construct Hamilton-Jacobi equations for a great variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical systems...). We recover all these, in principle, different cases using a unified framework based on skew-symmetric algebroids with a distinguished 1-cocycle. Several examples illustrate the theory.

Momentum and energy preserving integrators for nonholonomic dynamics

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian , where Q is the configuration manifold, and a (generally nonintegrable) distribution . In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study, in particular, the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.

On almost-Poisson structures in nonholonomic mechanics: II. The time-dependent framework

The structure of the equations of motion of a time-dependent mechanical system, subject to time-dependent non-holonomic constraints, is investigated in the Lagrangian as well as in the Hamiltonian setting. The treatment applies to systems with general nonlinear constraints, and the ambient space in which the constraint submanifold is embedded is equipped with a cosymplectic structure. In analogy with the autonomous case, it is shown that one can define an almost-Poisson structure on the constraint submanifold, which plays a prominent role in the description of non-holonomic dynamics. Moreover, it is seen that the corresponding almost-Poisson bracket can also be interpreted as a Dirac-type bracket. Systems with a Lagrangian of mechanical type and affine non-holonomic constraints are treated as a special case and two examples are discussed.