13,371 research outputs found
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
Tensor Integrals for Two Loop Standard Model Calculations
We give a new method for the reduction of tensor integrals to finite integral
representations and UV divergent analytic expressions. This includes a new
method for the handling of the gamma-algebra. TYPO IN EQUATION (5) CORRECTED,
MACROS REORDERED.Comment: 12 pages (LATEX
Discrete Routh Reduction
This paper develops the theory of abelian Routh reduction for discrete
mechanical systems and applies it to the variational integration of mechanical
systems with abelian symmetry. The reduction of variational Runge-Kutta
discretizations is considered, as well as the extent to which symmetry
reduction and discretization commute. These reduced methods allow the direct
simulation of dynamical features such as relative equilibria and relative
periodic orbits that can be obscured or difficult to identify in the unreduced
dynamics. The methods are demonstrated for the dynamics of an Earth orbiting
satellite with a non-spherical correction, as well as the double
spherical pendulum. The problem is interesting because in the unreduced
picture, geometric phases inherent in the model and those due to numerical
discretization can be hard to distinguish, but this issue does not appear in
the reduced algorithm, where one can directly observe interesting dynamical
structures in the reduced phase space (the cotangent bundle of shape space), in
which the geometric phases have been removed. The main feature of the double
spherical pendulum example is that it has a nontrivial magnetic term in its
reduced symplectic form. Our method is still efficient as it can directly
handle the essential non-canonical nature of the symplectic structure. In
contrast, a traditional symplectic method for canonical systems could require
repeated coordinate changes if one is evoking Darboux' theorem to transform the
symplectic structure into canonical form, thereby incurring additional
computational cost. Our method allows one to design reduced symplectic
integrators in a natural way, despite the noncanonical nature of the symplectic
structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added,
fixed typo
Deformations of vector-scalar models
Abelian vector fields non-minimally coupled to uncharged scalar fields arise
in many contexts. We investigate here through algebraic methods their
consistent deformations ("gaugings"), i.e., the deformations that preserve the
number (but not necessarily the form or the algebra) of the gauge symmetries.
Infinitesimal consistent deformations are given by the BRST cohomology classes
at ghost number zero. We parametrize explicitly these classes in terms of
various types of global symmetries and corresponding Noether currents through
the characteristic cohomology related to antifields and equations of motion.
The analysis applies to all ghost numbers and not just ghost number zero. We
also provide a systematic discussion of the linear and quadratic constraints on
these parameters that follow from higher-order consistency. Our work is
relevant to the gaugings of extended supergravities.Comment: v2: references added, typos corrected, minor changes for clarit
Bi-differential calculi and integrable models
The existence of an infinite set of conserved currents in completely
integrable classical models, including chiral and Toda models as well as the KP
and self-dual Yang-Mills equations, is traced back to a simple construction of
an infinite chain of closed (respectively, covariantly constant) 1-forms in a
(gauged) bi-differential calculus. The latter consists of a differential
algebra on which two differential maps act. In a gauged bi-differential
calculus these maps are extended to flat covariant derivatives.Comment: 24 pages, 2 figures, uses amssymb.sty and diagrams.sty, substantial
extensions of examples (relative to first version
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