A fully (3+1)-D Regge calculus model of the Kasner cosmology

We describe the first discrete-time 4-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of 4-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one parameter family of spacelike hypersurfaces built of tetrahedra. We implement a novel two-surface initial-data prescription for Regge calculus, and provide the first fully 4-dimensional application of an implicit decoupled evolution scheme (the ``Sorkin evolution scheme''). We benchmark this code on the Kasner cosmology --- a cosmology which embodies generic features of the collapse of many cosmological models. We (1) reproduce the continuum solution with a fractional error in the 3-volume of 10^{-5} after 10000 evolution steps, (2) demonstrate stable evolution, (3) preserve the standard deviation of spatial homogeneity to less than 10^{-10} and (4) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.Comment: 22 pages, 5 eps figures, LaTeX. Updated and expanded versio

Secondary Calculus and the Covariant Phase Space

The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of secondary calculus. In particular we describe the degeneracy distribution of w. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.Comment: 40 pages, typos correcte

Symbolic Computation of Variational Symmetries in Optimal Control

We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether's first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2,3,5,8).Comment: Presented at the 4th Junior European Meeting on "Control and Optimization", Bialystok Technical University, Bialystok, Poland, 11-14 September 2005. Accepted (24-Feb-2006) to Control & Cybernetic

Variational Sequences, Representation Sequences and Applications in Physics

This paper is a review containing new original results on the finite order variational sequence and its different representations with emphasis on applications in the theory of variational symmetries and conservation laws in physics

Coherent-State Approach to Two-dimensional Electron Magnetism

We study in this paper the possible occurrence of orbital magnetim for two-dimensional electrons confined by a harmonic potential in various regimes of temperature and magnetic field. Standard coherent state families are used for calculating symbols of various involved observables like thermodynamical potential, magnetic moment, or spatialdistribution of current. Their expressions are given in a closed form and the resulting Berezin-Lieb inequalities provide a straightforward way to study magnetism in various limit regimes. In particular, we predict a paramagnetic behaviour in the thermodynamical limit as well as in the quasiclassical limit under a weak field. Eventually, we obtain an exact expression for the magnetic moment which yields a full description of the phase diagram of the magnetization.Comment: 21 pages, 6 figures, submitted to PR

Variational Lie derivative and cohomology classes

We relate cohomology defined by a system of local Lagrangian with the cohomology class of the system of local variational Lie derivative, which is in turn a local variational problem; we show that the latter cohomology class is zero, since the variational Lie derivative `trivializes' cohomology classes defined by variational forms. As a consequence, conservation laws associated with symmetries ensuring the vanishing of the second variational derivative of a local variational problem are globally defined.Comment: 7 pages, misprints in Corollary 2 and a misleading in the abstract and the introduction corrected, XIX International Fall Workshop on Geometry and Physic