Geometric Mechanics and the Dynamics of Asteroid Pairs

This paper studies, using the technique of Lagrangian reduction, the geometric mechanics of a pair of asteroids in orbit about each other under mutual gravitational attraction

Cocycles, compatibility, and Poisson brackets for complex fluids

Motivated by Poisson structures for complex fluids containing cocycles, such as the Poisson structure for spin glasses given by Holm and Kupershmidt in 1988, we investigate a general construction of Poisson brackets with cocycles. Connections with the construction of compatible brackets found in the theory of integrable systems are also briefly discussed

Lagrangian Reduction by Stages

This booklet studies the geometry of the reduction of Lagrangian systems with symmetry in a way that allows the reduction process to be repeated; that is, it develops a context for Lagrangian reduction by stages. The Lagrangian reduction procedure focuses on the geometry of variational structures and how to reduce them to quotient spaces under group actions. This philosophy is well known for the classical cases, such as Routh reduction for systems with cyclic variables (where the symmetry group is Abelian) and Euler{Poincare reduction (for the case in which the conguration space is a Lie group) as well as Euler-Poincare reduction for semidirect products. The context established for this theory is a Lagrangian analogue of the bundle picture on the Hamiltonian side. In this picture, we develop a category that includes, as a special case, the realization of the quotient of a tangent bundle as the Whitney sum of the tangent of the quotient bundle with the associated adjoint bundle. The elements of this new category, called the Lagrange{Poincare category, have enough geometric structure so that the category is stable under the procedure of Lagrangian reduction. Thus, reduction may be repeated, giving the desired context for reduction by stages. Our category may be viewed as a Lagrangian analog of the category of Poisson manifolds in Hamiltonian theory. We also give an intrinsic and geometric way of writing the reduced equations, called the Lagrange{Poincare equations, using covariant derivatives and connections. In addition, the context includes the interpretation of cocycles as curvatures of connections and is general enough to encompass interesting situations involving both semidirect products and central extensions. Examples are given to illustrate the general theory. In classical Routh reduction one usually sets the conserved quantities conjugate to the cyclic variables equal to a constant. In our development, we do not require the imposition of this constraint. For the general theory along these lines, we refer to the complementary work of Marsden, Ratiu and Scheurle [2000], which studies the Lagrange-Routh equations

The Maxwell–Vlasov equations in Euler–Poincaré form

Low's well-known action principle for the Maxwell–Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamilton's principle for the Eulerian description of Low's action principle then casts the Maxwell–Vlasov equations into Euler–Poincaré form for right invariant motion on the diffeomorphism group of position-velocity phase space, [openface R]6. Legendre transforming the Eulerian form of Low's action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler–Poincaré equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie–Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell–Vlasov Poisson structure is known, whose ingredients are the Lie–Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born–Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin–Noether theorem for Euler–Poincaré equations and its meaning in the plasma context

An extension of the dirac and gotay-nester theories of constraints for dirac dynamical systems

This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. Integrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated. The cases considered usually in the literature correspond to a trivial foliation, with only one leaf. A Constraint Algorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an adapted total energy. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are, in a certain sense, dual and that using a combination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.Laboratorio de Electrónica Industrial, Control e Instrumentació

The Dirac theory of constraints, the Gotay-Nester theory and Poisson geometry

The Dirac theory of constraints has been widely studied and applied very successfully by physicists since the original works by Dirac and by Bergmann. From a mathematical standpoint, several aspects of the theory have been exposed rigorously afterwards by many authors. However, many questions related to, for instance, singular or infinite dimensional cases remain open. The work of Gotay and Nester presents a mathematical generalization in terms of presymplectic geometry, which introduces a dual point of view. We present a study of the Dirac theory of constraints emphasizing the duality between the Poisson-algebraic and the geometric points of view, related respectively to the work of Dirac and of Gotay and Nester, under strong regularity conditions. We deal with some questions insufficiently treated in the literature: a study of uniqueness of solution; avoiding almost completely the use of coordinates; the role of the Pontryagin bundle. We also show how one can globalize some results usually treated locally in the literature. For instance, we introduce the globalnotion of second class submanifoldas being tangent to a second class subbundle. A general study of global results for Dirac and Gotay-Nester theories remains an open question in this theory.La teoría de Dirac ha sido ampliamente estudiada y aplicada muy exitosamente por los físicos desde los trabajos originales de Dirac y de Bergmann. Desde un punto de vista matemático, varios aspectos de la teoría han sido expuestos rigurosamente por varios autores. Sin embargo, aún quedan abiertas varias preguntas relacionadas, por ejemplo, con casos singulares o infinito-dimensionales. El trabajo de Gotay y Nester presenta una generalización matemática en términos de la geometría presimpléctica, lo cual introduce un punto de vista dual. Presentamos un estudio de la teoría de ligaduras de Dirac enfatizando la dualidad entre los puntos de vista de las álgebras de Poisson y de la geometría presimpléctica, relacionados respectivamente con los trabajos de Dirac y de Gotay-Nester, bajo condiciones de regularidad fuertes. Abordamos algunas cuestiones insuficientemente tratadas en la literatura: un estudio de la unicidad de solución; evitar casi completamente el uso de coordenadas; el rol del fibrado de Pontryagin. También mostramos cómo se pueden globalizar algunos resultados usualmente tratados localmente en la literatura. Por ejemplo, introducimos la noción globalde subvariedad de segunda clasecomo variedad tangente a un subfibrado de segunda clase. Un estudio general de resultados globales para las teorías de Dirac y de Gotay-Nester sigue siendo una pregunta abierta en esta teoría.Instituto de Investigaciones en Electrónica, Control y Procesamiento de SeñalesFacultad de Ciencias Exacta

An extension of the dirac and gotay-nester theories of constraints for dirac dynamical systems

This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. Integrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated. The cases considered usually in the literature correspond to a trivial foliation, with only one leaf. A Constraint Algorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an adapted total energy. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are, in a certain sense, dual and that using a combination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.Laboratorio de Electrónica Industrial, Control e Instrumentació