Search

We express the first jet bundle of curves in Euclidean space as homogeneous spaces associated to a Galilean-type group. Certain Cartan connections on a manifold with values in the Lie algebra of the Galilean group are characterized as geometries associated to systems of second order ordinary differential equations. We show these Cartan connections admit a form of normal coordinates, and that in these normal coordinates the geodesic equations of the connection are second order ordinary differential equations. We then classify such connections by some of their torsions, extending a classical theorem of Chern involving the geometry associated to a system of second order differential equations., 6 pages, Latex2e

We show that every Lie algebra is equipped with a natural $(1,1)$-variant tensor field, the "canonical endomorphism field", naturally determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show its relevance for classical mechanics, in particular for Lax equations. We show that the space of Lax vector fields is closed under Lie bracket and we introduce a new bracket for vector fields on a Lie algebra. This bracket defines a new Lie structure on the space of vector fields., 18 pages

27 pages.-- MSC classes: 70F25; 37J60; 70G45., A k-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories., This work has been partially supported by MEC (Spain) Grant MTM 2007-62478, project “IngenioMathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM. Silvia Vilariño acknowledges the financial support of Xunta de Galicia Grants IN840C 2006/119-0 and IN809A 2007/151-0.

We introduce the notion of a symplectic Lie affgebroid and their Lagrangian submanifolds in order to describe the Lagrangian (Hamiltonian) dynamics on a Lie affgebroid in terms of this type of structures. Several examples are discussed., Comment: 50 pages. Several sections updated

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 as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems., This work has been partially supported by MICYT (Spain) Grants BMF 2003-01319, MTM 2004-7832 and BMF 2003-02532.

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., This work has been partially supported by MEC (Spain) Grants MTM 2004-7832, BFM2003- 01319 and BFM2003-02532.

Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which are closely related to affine analogs of Poisson and Jacobi structures. Homology and cohomology of the latter are canonically defined. The developed concepts are applied in solving some problems of frame-independent geometric description of mechanical systems., 37 pages, minor corrections, final version to appear in J. Geom. Phys

Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank two and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing two integrals of motion. We generalize the form of that Poisson structures and extend their domain of definition. We apply the theory to the rolling disk, the Routh's sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder., 22 pages

In this paper we discuss the relation between the unimodularity of a Lie algebroid $\tau_{A}: A\to Q$ and the existence of invariant volume forms for the hamiltonian dynamics on the dual bundle $A^{*}$. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids., Comment: 16 pages

In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry characterizing the different systems. In particular, we obtain that the class of the so-called algebroids covers a great variety of mechanical systems. Finally, as the main result, a hamiltonian symplectic realization of systems defined on algebroids is obtained.

In this paper we discuss variational constrained mechanics (vakonomic mechanics) on Lie affgebroids. We obtain the dynamical equations and the aff-Poisson bracket associated with a vakonomic system on a Lie affgebroid ${\mathcal A}$. We devote special attention to the particular case when the nonholonomic constraints are given by an affine subbundle of ${\mathcal A}$ and we discuss the variational character of the theory. Finally, we apply the results obtained to several examples., Comment: 23 pages

Using complexified quaternions, a formalism without Lorentz frames, and therefore also without vierbeins, for dealing with tensor and spinor fields in curved spacetime is presented. A local U(1) gauge symmetry, which, it is speculated, might be related to electromagnetism, emerges naturally., Comment: 14 pages; v2: minor corrections; v3: note added concerning unified treatment of local Lorentz transformations and local U(1) gauge transformations; v4: published in J. Math. Phys. 50 083507 (2009)

We start by formulating geometrically the Newton's law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. In fact, we use this scheme for further generalisation devoted to a constrained particle, to a discrete system of several free and constrained particles. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. In the second case, we obtain an explicit formula for the reaction force via the second fundamental form of the constrained configuration space. For multi--particle systems we describe geometrically the splitting related to the center of mass and relative velocities; in this way we emphasise the geometric source of classical formulas. Then, the above scheme is applied in detail to discrete rigid systems. We start by analysing the geometry of the rigid configuration space. In this way we recover the classical formula for the velocity of the rigid system via the parallelisation of Lie groups. Moreover, we study in detail the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields an explicit formula for the reaction force in the nodes of the rigid constraint., Comment: 48 pages, no figures, style files included in the source file

We propose a suitable formulation of the Hamiltonian formalism for Field Theory in terms of Hamiltonian connections and multisymplectic forms where a composite fibered bundle, involving a line bundle, plays the role of an extended configuration bundle. This new approach can be interpreted as a suitable generalization to Field Theory of the homogeneous formalism for Hamiltonian Mechanics. As an example of application, we obtain the expression of a formal energy for a parametrized version of the Hilbert--Einstein Lagrangian and we show that this quantity is conserved., 9 pages, slightly revised, to appear in Proc. Winter School "Geometry and Physics", Srni (CZ) 2006

Traditionally, the discussion about the geometrical interpretation of inertial forces is reserved for General Relativity handbooks. In these notes an analysis of the effect of such forces in a classical (newtonian) context is made, as well as a study of the relation between the precession of the Foucault pendulum and the holonomy on a surface, including some critical remarks about the common statement \textquotedblleft precession of Foucault pendulum equals holonomy on the sphere\textquotedblright., Some typos corrected

The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain one-dimensional complex vector bundle over space-time. This bundle, constructed from the data provided by all possible inertial observers, has no canonical trivialization, so these sections cannot be viewed as functions on the space-time. The presented framework is conceptually four-dimensional and does not involve any ad hoc or axiomatically introduced geometrical structures. It is based only on the traditional understanding of the Schroedinger operator in a given reference frame and it turns out to be strictly related to the frame-independent formulation of analytical Newtonian mechanics that makes a bridge between the classical and quantum theory., Comment: 7 pages

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler-Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem. Besides, the constructions seem to be more natural and simpler., 13 pages, minor changes, the version to appear in Int. J. Geom. Meth. Mod. Phys

A frame independent formulation of analytical mechanics in the Newtonian space-time is presented The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is uded. Lagragian and hamiltonian generating objects, together with the Legendre transformation independent on inertial frame are constructed., Comment: Revised and corrected, one subsection added. 16 pages, accepted in ROMP

We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in particular, related to geometrization of classical and quantum mechanical and field theories, even if we restrict our considerations only to commutative and noncommutative Riemannian spaces. We elaborate a geometric approach to the Clifford modules adapted to nonlinear connections, to the theory of spinors and the Dirac operators on nonholonomic spaces and consider possible generalizations to noncommutative geometry. We argue that any commutative Riemann--Finsler geometry and generalizations my be derived from noncommutative geometry by applying certain methods elaborated for Riemannian spaces but extended to nonholonomic frame transforms and manifolds provided with nonlinear connection structure., Comment: 55 pages, latex 2e, Outline of a series of Lectures and Seminars

A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is used. Lagrangian and hamiltonian generating objects, together with the Legendre transformation independent on inertial frame are constructed., Comment: 19 pages

It is shown that in the semi-classical approximation of the electroweak sector of the Standard Model the moduli space of vacua can be identified with the first de Rham cohomology group of space-time. This gives a slightly different physical interpretation of the occurrence of the well-known Ahoronov-Bohm effect. Moreover, when charge conjugation is taken into account, the existence of a non-trivial ground state of the Higgs boson is shown to be equivalent to the triviality of the electroweak gauge bundle. As a consequence, the gauge bundle of the electromagnetic interaction must also be trivial. Though derived at ``tree level'' the results presented here may also have some consequences for quantizing, e. g., electromagnetism on an arbitrary curved space-time., Comment: 26 pages, no figures

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this case some part of normality equations describe generalized Legendre transformation itself irrespective to that Newtonian dynamical system, for which others are written. In present paper this smaller part of normality equations is studied., AmSTeX, 19 pages, amsppt style, 2 figures in 1 gif-file

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by classical and/or generalized Legendre transformation. However, in this case they appear to be written in p-representation, i. e. in terms of momentum covector and its components. The goal of present paper is to transform normality equations back to v-representation, which is more natural for Newtonian dynamical systems., AmSTeX, 32 pages, amsppt style

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_VY$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field momentum observables on $Z$ are generalized by those on $L_VY$. Furthermore, momentum observables on $L_VY$ produce conserved quantities along flows in $L_VY$. For translational and orthogonal symmetries of fields and reparametrization symmetry in mechanics, momentum is conserved, and for angular momentum in time-evolution mechanics we produce a version of the parallel axis theorem of rotational dynamics, and in special relativity, we produce the transformation of angular momentum under boosts., 23 pages