884 research outputs found
The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions
We consider the problem of integrability of the Poisson equations describing
spatial motion of a rigid body in the classical nonholonomic Suslov problem. We
obtain necessary conditions for their solutions to be meromorphic and show that
under some further restrictions these conditions are also sufficient. The
latter lead to a family of explicit meromorphic solutions, which correspond to
rather special motions of the body in space. We also give explicit extra
polynomial integrals in this case.
In the more general case (but under one restriction), the Poisson equations
are transformed into a generalized third order hypergeometric equation. A study
of its monodromy group allows us also to calculate the "scattering" angle: the
angle between the axes of limit permanent rotations of the body in space
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
RG transport theory for open quantum systems: Charge fluctuations in multilevel quantum dots in and out of equilibrium
We present the real-time renormalization group (RTRG) method as a method to
describe the stationary state current through generic multi-level quantum dots
with a complex setup in nonequilibrium. The employed approach consists of a
very rudiment approximation for the RG equations which neglects all vertex
corrections while it provides a means to compute the effective dot Liouvillian
self-consistently. Being based on a weak-coupling expansion in the tunneling
between dot and reservoirs, the RTRG approach turns out to reliably describe
charge fluctuations in and out of equilibrium for arbitrary coupling strength,
even at zero temperature. We confirm this in the linear response regime with a
benchmark against highly-accurate numerically renormalization group data in the
exemplary case of three-level quantum dots. For small to intermediate bias
voltages and weak Coulomb interactions, we find an excellent agreement between
RTRG and functional renormalization group data, which can be expected to be
accurate in this regime. As a consequence, we advertise the presented RTRG
approach as an efficient and versatile tool to describe charge fluctuations
theoretically in quantum dot systems
Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields
In this paper we analyse the integrability of a dynamical system describing
the rotational motion of a rigid satellite under the influence of gravitational
and magnetic fields. In our investigations we apply an extension of the Ziglin
theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric
satellite the system does not admit an additional real meromorphic first
integral except for one case when the value of the induced magnetic moment
along the symmetry axis is related to the principal moments of inertia in a
special way.Comment: 39 pages, 4 figures, missing bibliography was adde
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