David Martín de Diego

On the geometry of non‐holonomic Lagrangian systems

We present a geometric framework for non‐holonomic Lagrangian systems in terms of distributions on the configuration manifold. If the constrained system is regular, an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics. If the constrained system is singular, we develop a constraint algorithm which is very similar to that developed by Dirac and Bergmann, and later globalized by Gotay and Nester. Special attention to the case of constrained systems given by connections is paid. In particular, we extend the results of Koiller for Caplygin systems. An application to the so‐called non‐holonomic geometry is given.

Dynamics of generalized Poisson and Nambu–Poisson brackets

A unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.

Non-holonomic Lagrangian systems in jet manifolds

A geometrical setting in terms of jet manifolds is developed for time-dependent non-holonomic Lagrangian systems. An almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics. A constrained Poincare - Cartan 2-form is defined. If the non-holonomic system is singular, a constraint algorithm is constructed. Special attention is devoted to Caplygin systems and a reduction theorem is proved.

Reduction of nonholonomic mechanical systems with symmetries

Abstract A geometric reduction procedure is presented for Lagrangian systems subjected to nonlinear nonholonomic constraints in the presence of symmetries. Our approach is based on a geometrical method which enables one to deduce the constrained dynamics from the unconstrained one by projection.

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 Caplygin systems.

Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. From a variational principle we derive the discrete Euler–Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include the classical discrete Euler–Lagrange equations, the discrete Euler–Poincare and discrete Lagrange–Poincare equations as particular cases. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.

Mechanical systems with nonlinear constraints

A geometrical formalism for nonlinear nonholonomic Lagrangian systems is developed. The solution of the constrained problem is discussed by using almost product structures along the constraint submanifold. Constrained systems with ideal constraints are also considered, and Chetaev conditions are given in geometrical terms. A Noether theorem is also proved.

Dirac brackets in constrained dynamics

This work has been partially supported through grants DGICYT (Spain) (Projects PB94-0106, PB97- 1487 and PB95-0401), ConsejeroÂa de EducacioÂn y Cultura de la Comunidad AutoÂnoma de Madrid, and UNED (Spain)

A generalization of Chetaev’s principle for a class of higher order nonholonomic constraints

The constraint distribution in nonholonomic mechanics has a double role. On the one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D’Alembert’s principle to derive the equations of motion. We will show that many systems of physical interest where D’Alembert’s principle does not apply can be conveniently modeled within the general idea of the principle of virtual work by the introduction of both kinematic constraints and variational constraints as being independent entities. This includes, for example, elastic rolling bodies and pneumatic tires. Also, D’Alembert’s principle and Chetaev’s principle fall into this scheme. We emphasize the geometric point of view, avoiding the use of local coordinates, which is the appropriate setting for dealing with questions of global nature, like reduction.

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

Abstract This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic discrete Euler–Lagrange equations in a setting that permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot).