368 research outputs found
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.Comment: 29 pages; typos remove
Discrete Nonholonomic Lagrangian Systems on Lie Groupoids
This paper studies the construction of geometric integrators for nonholonomic
systems. We derive the nonholonomic discrete Euler-Lagrange equations in a
setting which 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).Comment: 45 page
Complexification of Gauge Theories
For the case of a first-class constrained system with an equivariant momentum
map, we study the conditions under which the double process of reducing to the
constraint surface and dividing out by the group of gauge transformations
is equivalent to the single process of dividing out the initial phase space by
the complexification of . For the particular case of a phase space
action that is the lift of a configuration space action, conditions are found
under which, in finite dimensions, the physical phase space of a gauge system
with first-class constraints is diffeomorphic to a manifold imbedded in the
physical configuration space of the complexified gauge system. Similar
conditions are shown to hold in the infinite-dimensional example of Yang-Mills
theories. As a physical application we discuss the adequateness of using
holomorphic Wilson loop variables as (generalized) global coordinates on the
physical phase space of Yang-Mills theory.Comment: 25pp., LaTeX, Syracuse SU-GP-93/6-2, Lisbon DF/IST 6.9
The general (2,2) gauged sigma model with three--form flux
We find the conditions under which a Riemannian manifold equipped with a
closed three-form and a vector field define an on--shell N=(2,2) supersymmetric
gauged sigma model. The conditions are that the manifold admits a twisted
generalized Kaehler structure, that the vector field preserves this structure,
and that a so--called generalized moment map exists for it. By a theorem in
generalized complex geometry, these conditions imply that the quotient is again
a twisted generalized Kaehler manifold; this is in perfect agreement with
expectations from the renormalization group flow. This method can produce new
N=(2,2) models with NS flux, extending the usual Kaehler quotient construction
based on Kaehler gauged sigma models.Comment: 24 pages. v2: typos fixed, other minor correction
Recent progress on univariate and multivariate polynomial and spline quasi-interpolants
Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral. We shall also present some applications of QIs to numerical methods
Observers in an accelerated universe
If the current acceleration of our Universe is due to a cosmological
constant, then a Coleman-De Luccia bubble will nucleate in our Universe. In
this work, we consider that our observations could be likely in this framework,
consisting in two infinite spaces, if a foliation by constant mean curvature
hypersurfaces is taken to count the events in the spacetime. Thus, we obtain
and study a particular foliation, which covers the existence of most observers
in our part of spacetime.Comment: revised version, accepted in EPJ
On integrability of Hirota-Kimura type discretizations
We give an overview of the integrability of the Hirota-Kimura discretization
method applied to algebraically completely integrable (a.c.i.) systems with
quadratic vector fields. Along with the description of the basic mechanism of
integrability (Hirota-Kimura bases), we provide the reader with a fairly
complete list of the currently available results for concrete a.c.i. systems.Comment: 47 pages, some minor change
Invariant higher-order variational problems II
Motivated by applications in computational anatomy, we consider a
second-order problem in the calculus of variations on object manifolds that are
acted upon by Lie groups of smooth invertible transformations. This problem
leads to solution curves known as Riemannian cubics on object manifolds that
are endowed with normal metrics. The prime examples of such object manifolds
are the symmetric spaces. We characterize the class of cubics on object
manifolds that can be lifted horizontally to cubics on the group of
transformations. Conversely, we show that certain types of non-horizontal
geodesics on the group of transformations project to cubics. Finally, we apply
second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics
on the group of transformations. This leads to a reduced form of the equations
that reveals the obstruction for the projection of a cubic on a transformation
group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome
Foliations of Isonergy Surfaces and Singularities of Curves
It is well known that changes in the Liouville foliations of the isoenergy
surfaces of an integrable system imply that the bifurcation set has
singularities at the corresponding energy level. We formulate certain
genericity assumptions for two degrees of freedom integrable systems and we
prove the opposite statement: the essential critical points of the bifurcation
set appear only if the Liouville foliations of the isoenergy surfaces change at
the corresponding energy levels. Along the proof, we give full classification
of the structure of the isoenergy surfaces near the critical set under our
genericity assumptions and we give their complete list using Fomenko graphs.
This may be viewed as a step towards completing the Smale program for relating
the energy surfaces foliation structure to singularities of the momentum
mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure
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