5,001 research outputs found
On numerical approaches to the analysis of topology of the phase space for dynamical integrability
In this paper we consider the possibility to use numerical simulations for a
computer assisted analysis of integrability of dynamical systems. We formulate
a rather general method of recovering the obstruction to integrability for the
systems with a small number of degrees of freedom. We generalize this method
using the results of KAM theory and stochastic approaches to the families of
parameter depending systems. This permits the localization of possible
integrability regions in the parameter space. We give some examples of
application of this approach to dynamical systems having mechanical origin.Comment: 9 figures, version accepted to CS
Graded geometry in gauge theories and beyond
We study some graded geometric constructions appearing naturally in the
context of gauge theories. Inspired by a known relation of gauging with
equivariant cohomology we generalize the latter notion to the case of arbitrary
Q-manifolds introducing thus the concept of equivariant Q-cohomology. Using
this concept we describe a procedure for analysis of gauge symmetries of given
functionals as well as for constructing functionals (sigma models) invariant
under an action of some gauge group.
As the main example of application of these constructions we consider the
twisted Poisson sigma model. We obtain it by a gauging-type procedure of the
action of an essentially infinite dimensional group and describe its symmetries
in terms of classical differential geometry.
We comment on other possible applications of the described concept including
the analysis of supersymmetric gauge theories and higher structures.Comment: version accepted to Journal of Geometry and Physics, updated
reference
Dirac Sigma Models from Gauging
The G/G WZW model results from the WZW-model by a standard procedure of
gauging. G/G WZW models are members of Dirac sigma models, which also contain
twisted Poisson sigma models as other examples. We show how the general class
of Dirac sigma models can be obtained from a gauging procedure adapted to Lie
algebroids in the form of an equivariantly closed extension. The rigid gauge
groups are generically infinite dimensional and a standard gauging procedure
would give a likewise infinite number of 1-form gauge fields; the proposed
construction yields the requested finite number of them.
Although physics terminology is used, the presentation is kept accessible
also for a mathematical audience.Comment: 20 pages, 3 figure
Effective algorithm of analysis of integrability via the Ziglin's method
In this paper we continue the description of the possibilities to use
numerical simulations for mathematically rigorous computer assisted analysis of
integrability of dynamical systems. We sketch some of the algebraic methods of
studying the integrability and present a constructive algorithm issued from the
Ziglin's approach. We provide some examples of successful applications of the
constructed algorithm to physical systems.Comment: a figure added, version accepted to JDC
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