554 research outputs found
Integrable and superintegrable systems with spin
A system of two particles with spin s=0 and s=1/2 respectively, moving in a
plane is considered. It is shown that such a system with a nontrivial
spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order
integrals of motion. The Pauli equation is solved in this superintegrable case
and reduced to a system of ordinary differential equations when only one
first-order integral exists.Comment: 12 page
Addition theorems and the Drach superintegrable systems
We propose new construction of the polynomial integrals of motion related to
the addition theorems. As an example we reconstruct Drach systems and get some
new two-dimensional superintegrable Stackel systems with third, fifth and
seventh order integrals of motion.Comment: 18 pages, the talk given on the conference "Superintegrable Systems
in Classical and Quantum Mechanics", Prague 200
Third-order superintegrable systems separable in parabolic coordinates
In this paper, we investigate superintegrable systems which separate in
parabolic coordinates and admit a third-order integral of motion. We give the
corresponding determining equations and show that all such systems are
multi-separable and so admit two second-order integrals. The third-order
integral is their Lie or Poisson commutator. We discuss how this situation is
different from the Cartesian and polar cases where new potentials were
discovered which are not multi-separable and which are expressed in terms of
Painlev\'e transcendents or elliptic functions
ANALYSIS OF WALL SHAPE IN INDOOR AIR CIRCULATION BY THE FINITE ELEMENT METHOD
The use of computational models in built environments comes from the need to deal with situations as close as possible to the reality and also to study functional spaces that could be able to provide, for example, thermal comfort. In this work we analyze some cases of indoor air circulation in built environments through a mixed stabilized finite element method, applied to the Navier-Stokes equations in velocity and pressure variables. The implemented numerical method ensures stability for the internal constraint imposed by the velocity field, and accommodates moderate to large advective effects. The obtained internal wind field allows the choice of wall shapes that increase or not the ventilation and can alter its distribution, allowing in this way a better adequacy of the built environment for the climate needs and its objectives
On the Drach superintegrable systems
Cubic invariants for two-dimensional degenerate Hamiltonian systems are
considered by using variables of separation of the associated St\"ackel
problems with quadratic integrals of motion. For the superintegrable St\"ackel
systems the cubic invariant is shown to admit new algebro-geometric
representation that is far more elementary than the all the known
representations in physical variables. A complete list of all known systems on
the plane which admit a cubic invariant is discussed.Comment: 16 pages, Latex2e+Amssym
Superintegrable Systems with a Third Order Integrals of Motion
Two-dimensional superintegrable systems with one third order and one lower
order integral of motion are reviewed. The fact that Hamiltonian systems with
higher order integrals of motion are not the same in classical and quantum
mechanics is stressed. New results on the use of classical and quantum third
order integrals are presented in Section 5 and 6.Comment: To appear in J. Phys A: Mathematical and Theoretical (SPE QTS5
Hamiltonians separable in cartesian coordinates and third-order integrals of motion
We present in this article all Hamiltonian systems in E(2) that are separable
in cartesian coordinates and that admit a third-order integral, both in quantum
and in classical mechanics. Many of these superintegrable systems are new, and
it is seen that there exists a relation between quantum superintegrable
potentials, invariant solutions of the Korteweg-De Vries equation and the
Painlev\'e transcendents.Comment: 19 pages, Will be published in J. Math. Phy
Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials
We consider a superintegrable quantum potential in two-dimensional Euclidean
space with a second and a third order integral of motion. The potential is
written in terms of the fourth Painleve transcendent. We construct for this
system a cubic algebra of integrals of motion. The algebra is realized in terms
of parafermionic operators and we present Fock type representations which yield
the corresponding energy spectra. We also discuss this potential from the point
of view of higher order supersymmetric quantum mechanics and obtain ground
state wave functions.Comment: 26 page
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