5,005 research outputs found
Finding Liouvillian first integrals of rational ODEs of any order in finite terms
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher
and others, that if a given rational ODE has a Liouvillian first integral then
the corresponding integrating factor of the ODE must be of a very special form
of a product of powers and exponents of irreducible polynomials. These results
lead to a partial algorithm for finding Liouvillian first integrals. However,
there are two main complications on the way to obtaining polynomials in the
integrating factor form. First of all, one has to find an upper bound for the
degrees of the polynomials in the product above, an unsolved problem, and then
the set of coefficients for each of the polynomials by the
computationally-intensive method of undetermined parameters. As a result, this
approach was implemented in CAS only for first and relatively simple second
order ODEs. We propose an algebraic method for finding polynomials of the
integrating factors for rational ODEs of any order, based on examination of the
resultants of the polynomials in the numerator and the denominator of the
right-hand side of such equation. If both the numerator and the denominator of
the right-hand side of such ODE are not constants, the method can determine in
finite terms an explicit expression of an integrating factor if the ODE permits
integrating factors of the above mentioned form and then the Liouvillian first
integral. The tests of this procedure based on the proposed method, implemented
in Maple in the case of rational integrating factors, confirm the consistence
and efficiency of the method.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Two-dimensional metric-affine gravity
There is a number of completely integrable gravity theories in two
dimensions. We study the metric-affine approach on a 2-dimensional spacetime
and display a new integrable model. Its properties are described and compared
with the known results of Poincare gauge gravity.Comment: Revtex, 15 pages, no figure
Plane waves in metric-affine gravity
We describe plane-fronted waves in the Yang-Mills type quadratic
metric-affine theory of gravity. The torsion and the nonmetricity are both
nontrivial, and they do not belong to the triplet ansatz.Comment: 18 pages, revtex, no figure
Exact Solutions in Poincar\'e Gauge Gravity Theory
In the framework of the gauge theory based on the Poincar\'e symmetry group,
the gravitational field is described in terms of the coframe and the local
Lorentz connection. Considered as gauge field potentials, they give rise to the
corresponding field strength which are naturally identified with the torsion
and the curvature on the Riemann--Cartan spacetime. We study the class of
quadratic Poincar\'e gauge gravity models with the most general Yang--Mills
type Lagrangian which contains all possible parity-even and parity-odd
invariants built from the torsion and the curvature. Exact vacuum solutions of
the gravitational field equations are constructed as a certain deformation of
de Sitter geometry. They are black holes with nontrivial torsion.Comment: 17 pages, no figure
Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)
We show that the -dimensional Euler--Manakov top on can be
represented as a Poisson reduction of an integrable Hamiltonian system on a
symplectic extended Stiefel variety , and present its Lax
representation with a rational parameter.
We also describe an integrable two-valued symplectic map on the
4-dimensional variety . The map admits two different reductions,
namely, to the Lie group SO(3) and to the coalgebra .
The first reduction provides a discretization of the motion of the classical
Euler top in space and has a transparent geometric interpretation, which can be
regarded as a discrete version of the celebrated Poinsot model of motion and
which inherits some properties of another discrete system, the elliptic
billiard.
The reduction of to gives a new explicit discretization of
the Euler top in the angular momentum space, which preserves first integrals of
the continuous system.Comment: 18 pages, 1 Figur
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