10,766 research outputs found
Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform
This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different spaces. Through the use of this tool we derive and classify for the first time all two-dimensional (2D) superintegrable systems. The underlying spaces are exactly those derived by Koenigs in his remarkable paper giving all 2D manifolds (with zero potential) that admit at least three second order symmetries. Our derivation is very simple and quite distinct. We also show that every superintegrable system is the Stäckel transform of a superintegrable system on a constant curvature space
Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems
This paper is the conclusion of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. For two-dimensional and for conformally flat three-dimensional spaces with nondegenerate potentials we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension. We also correct an error in an earlier paper in the series (that does not alter the structure results) and we elucidate the distinction between superintegrable systems with bases of functionally linearly independent and functionally linearly dependent symmetries
Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory
This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally flat spaces. Here we consider classical superintegrable systems with nondegenerate potentials in three dimensions. We show that there exists a standard structure for such systems, based on the algebra of 3×3 symmetric matrices, and that the quadratic algebra always closes at order 6. We show that the spaces of truly second-, third-, fourth-, and sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively, and we construct explicit bases for the fourth- and sixth order constants in terms of products of the second order constants
Tools for Verifying Classical and Quantum Superintegrability
Recently many new classes of integrable systems in n dimensions occurring in
classical and quantum mechanics have been shown to admit a functionally
independent set of 2n-1 symmetries polynomial in the canonical momenta, so that
they are in fact superintegrable. These newly discovered systems are all
separable in some coordinate system and, typically, they depend on one or more
parameters in such a way that the system is superintegrable exactly when some
of the parameters are rational numbers. Most of the constructions to date are
for n=2 but cases where n>2 are multiplying rapidly. In this article we
organize a large class of such systems, many new, and emphasize the underlying
mechanisms which enable this phenomena to occur and to prove
superintegrability. In addition to proofs of classical superintegrability we
show that the 2D caged anisotropic oscillator and a Stackel transformed version
on the 2-sheet hyperboloid are quantum superintegrable for all rational
relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum
superintegrable for all rational values of a parameter k in the potential
Fatigue Damage in Notched Composite Laminates Under Tension-Tension Cyclic Loads
The results are given of an investigation to determine the damage states which develop in graphite epoxy laminates with center holes due to tension-tension cyclic loads, to determine the influence of stacking sequence on the initiation and interaction of damage modes and the process of damage development, and to establish the relationships between the damage states and the strength, stiffness, and life of the laminates. Two quasi-isotropic laminates were selected to give different distributions of interlaminar stresses around the hole. The laminates were tested under cyclic loads (R=0.1, 10 Hz) at maximum stresses ranging between 60 and 95 percent of the notched tensile strength
Superintegrable Systems in Darboux spaces
Almost all research on superintegrable potentials concerns spaces of constant
curvature. In this paper we find by exhaustive calculation, all superintegrable
potentials in the four Darboux spaces of revolution that have at least two
integrals of motion quadratic in the momenta, in addition to the Hamiltonian.
These are two-dimensional spaces of nonconstant curvature. It turns out that
all of these potentials are equivalent to superintegrable potentials in complex
Euclidean 2-space or on the complex 2-sphere, via "coupling constant
metamorphosis" (or equivalently, via Staeckel multiplier transformations). We
present tables of the results
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