72,952 research outputs found

### New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory

DifferentialGeometry is a Maple software package which symbolically performs
fundamental operations of calculus on manifolds, differential geometry, tensor
calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the
variational calculus. These capabilities, combined with dramatic recent
improvements in symbolic approaches to solving algebraic and differential
equations, have allowed for development of powerful new tools for solving
research problems in gravitation and field theory. The purpose of this paper is
to describe some of these new tools and present some advanced applications
involving: Killing vector fields and isometry groups, Killing tensors and other
tensorial invariants, algebraic classification of curvature, and symmetry
reduction of field equations.Comment: 42 page

### Symmetries of the Einstein Equations

Generalized symmetries of the Einstein equations are infinitesimal
transformations of the spacetime metric that formally map solutions of the
Einstein equations to other solutions. The infinitesimal generators of these
symmetries are assumed to be local, \ie at a given spacetime point they are
functions of the metric and an arbitrary but finite number of derivatives of
the metric at the point. We classify all generalized symmetries of the vacuum
Einstein equations in four spacetime dimensions and find that the only
generalized symmetry transformations consist of: (i) constant scalings of the
metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms.
Our results rule out a large class of possible ``observables'' for the
gravitational field, and suggest that the vacuum Einstein equations are not
integrable.Comment: 15 pages, FTG-114-USU, Plain Te

### Properties of the Scalar Universal Equations

The variational properties of the scalar so--called ``Universal'' equations
are reviewed and generalised. In particular, we note that contrary to earlier
claims, each member of the Euler hierarchy may have an explicit field
dependence. The Euler hierarchy itself is given a new interpretation in terms
of the formal complex of variational calculus, and is shown to be related to
the algebra of distinguished symmetries of the first source form.Comment: 15 pages, LaTeX articl

### Presymplectic current and the inverse problem of the calculus of variations

The inverse problem of the calculus of variations asks whether a given system
of partial differential equations (PDEs) admits a variational formulation. We
show that the existence of a presymplectic form in the variational bicomplex,
when horizontally closed on solutions, allows us to construct a variational
formulation for a subsystem of the given PDE. No constraints on the
differential order or number of dependent or independent variables are assumed.
The proof follows a recent observation of Bridges, Hydon and Lawson and
generalizes an older result of Henneaux from ordinary differential equations
(ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.Comment: v2: 17 pages, no figures, BibTeX; minor corrections, close to
published versio

### On a Order Reduction Theorem in the Lagrangian Formalism

We provide a new proof of a important theorem in the Lagrangian formalism
about necessary and sufficient conditions for a second-order variational system
of equations to follow from a first-order Lagrangian.Comment: 9 pages, LATEX, no figures; appear in Il Nuovo Cimento

### Superposition Formulas for Darboux Integrable Exterior Differential Systems

In this paper we present a far-reaching generalization of E. Vessiot's
analysis of the Darboux integrable partial differential equations in one
dependent and two independent variables. Our approach provides new insights
into this classical method, uncovers the fundamental geometric invariants of
Darboux integrable systems, and provides for systematic, algorithmic
integration of such systems. This work is formulated within the general
framework of Pfaffian exterior differential systems and, as such, has
applications well beyond those currently found in the literature. In
particular, our integration method is applicable to systems of hyperbolic PDE
such as the Toda lattice equations, 2 dimensional wave maps and systems of
overdetermined PDE.Comment: 80 page report. Updated version with some new sections, and major
improvements to other

### The Principle of Symmetric Criticality in General Relativity

We consider a version of Palais' Principle of Symmetric Criticality (PSC)
that is applicable to the Lie symmetry reduction of Lagrangian field theories.
PSC asserts that, given a group action, for any group-invariant Lagrangian the
equations obtained by restriction of Euler-Lagrange equations to
group-invariant fields are equivalent to the Euler-Lagrange equations of a
canonically defined, symmetry-reduced Lagrangian. We investigate the validity
of PSC for local gravitational theories built from a metric. It is shown that
there are two independent conditions which must be satisfied for PSC to be
valid. One of these conditions, obtained previously in the context of
transverse symmetry group actions, provides a generalization of the well-known
unimodularity condition that arises in spatially homogeneous cosmological
models. The other condition seems to be new. The conditions that determine the
validity of PSC are equivalent to pointwise conditions on the group action
alone. These results are illustrated with a variety of examples from general
relativity. It is straightforward to generalize all of our results to any
relativistic field theory.Comment: 46 pages, Plain TeX, references added in revised versio

### Magnetically generated spin-orbit coupling for ultracold atoms

We present a new technique for producing two- and three-dimensional
Rashba-type spin-orbit couplings for ultracold atoms without involving light.
The method relies on a sequence of pulsed inhomogeneous magnetic fields
imprinting suitable phase gradients on the atoms. For sufficiently short pulse
durations, the time-averaged Hamiltonian well approximates the Rashba
Hamiltonian. Higher order corrections to the energy spectrum are calculated
exactly for spin-1/2 and perturbatively for higher spins. The pulse sequence
does not modify the form of rotationally symmetric atom-atom interactions.
Finally, we present a straightforward implementation of this pulse sequence on
an atom chip

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