4,360 research outputs found
Symmetry Properties of Autonomous Integrating Factors
We study the symmetry properties of autonomous integrating factors from an
algebraic point of view. The symmetries are delineated for the resulting
integrals treated as equations and symmetries of the integrals treated as
functions or configurational invariants. The succession of terms (pattern) is
noted. The general pattern for the solution symmetries for equations in the
simplest form of maximal order is given and the properties of the associated
integrals resulting from this analysis are given.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Gauge Variant Symmetries for the Schr\"odinger Equation
The last multiplier of Jacobi provides a route for the determination of
families of Lagrangians for a given system. We show that the members of a
family are equivalent in that they differ by a total time derivative. We derive
the Schr\"odinger equation for a one-degree-of-freedom system with a constant
multiplier. In the sequel we consider the particular example of the simple
harmonic oscillator. In the case of the general equation for the simple
harmonic oscillator which contains an arbitrary function we show that all
Schr\"odinger equations possess the same number of Lie point symmetries with
the same algebra. From the symmetries we construct the solutions of the
Schr\"odinger equation and find that they differ only by a phase determined by
the gauge.Comment: 12 page
Integrability of the Einstein-nonlinear -model in a nontrivial topological sector
The integrability of the\ Einstein-nonlinear -model
with nonvanishing cosmological charge is studied. We apply the method of
singularity analysis of differential equations and we show that the equations
for the gravitational field are integrable. The first few terms of the solution
are presented.Comment: 6 pages, 2 figures, published at EPJ
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