1,107 research outputs found
A New Superintegrable Hamiltonian
We identify a new superintegrable Hamiltonian in 3 degrees of freedom,
obtained as a reduction of pure Keplerian motion in 6 dimensions. The new
Hamiltonian is a generalization of the Keplerian one, and has the familiar 1/r
potential with three barrier terms preventing the particle crossing the
principal planes. In 3 degrees of freedom, there are 5 functionally independent
integrals of motion, and all bound, classical trajectories are closed and
strictly periodic. The generalisation of the Laplace-Runge-Lenz vector is
identified and shown to provide functionally independent isolating integrals.
They are quartic in the momenta and do not arise from separability of the
Hamilton-Jacobi equation. A formulation of the system in action-angle variables
is presented.Comment: 11 pages, 4 figures, submitted to The Journal of Mathematical Physic
Second order superintegrable systems in conformally flat spaces. IV. The classical 3D StÀckel transform and 3D classification theory
This article 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. In the first part of the article we study the StÀckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different three-dimensional spaces. We show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is StÀckel equivalent to a system on a constant curvature space. In the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates. We find that there are eight families of these systems
Families of classical subgroup separable superintegrable systems
We describe a method for determining a complete set of integrals for a
classical Hamiltonian that separates in orthogonal subgroup coordinates. As
examples, we use it to determine complete sets of integrals, polynomial in the
momenta, for some families of generalized oscillator and Kepler-Coulomb
systems, hence demonstrating their superintegrability. The latter generalizes
recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz.
Another example is given of a superintegrable system on a non-conformally flat
space.Comment: 9 page
Young People as Humans in Family Court Processes: A Child Rights Approach to Legal Representation
The authors, a retired British Columbia Supreme Court judge and a senior member of Ontarioâs Office of the Childrenâs Lawyer, address the important issue of legal representation for children. They are co-chairs of the Steering Committee which guided the development of the Canadian Bar Associationâs new and comprehensive Child Rights Toolkit. As such, they are well-placed to discuss how a child rights approach, as required by the United Nations Convention on the Rights of the Child to which Canada is a ratifying party, supports legal representation for children who find themselves caught in contentious family law proceedings before the courts
Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability
The existence of a Lagrangian description for the second-order Riccati
equation is analyzed and the results are applied to the study of two different
nonlinear systems both related with the generalized Riccati equation. The
Lagrangians are nonnatural and the forces are not derivable from a potential.
The constant value of a preserved energy function can be used as an
appropriate parameter for characterizing the behaviour of the solutions of
these two systems. In the second part the existence of two--dimensional
versions endowed with superintegrability is proved. The explicit expressions of
the additional integrals are obtained in both cases. Finally it is proved that
the orbits of the second system, that represents a nonlinear oscillator, can be
considered as nonlinear Lissajous figuresComment: 25 pages, 7 figure
Superintegrability in a two-dimensional space of nonconstant curvature
A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated HamiltonâJacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined
Quantum models related to fouled Hamiltonians of the harmonic oscillator
We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator
which provide, at the classical level, the same equation of motion as the
conventional Hamiltonian. These Hamiltonians, say and , result
to be explicitly time-dependent and can be expressed as a formal rotation of
two cubic polynomial functions, and , of the canonical variables
(q,p).
We investigate the role of these fouled Hamiltonians at the quantum level.
Adopting a canonical quantization procedure, we construct some quantum models
and analyze the related eigenvalue equations. One of these models is described
by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a
discrete spectrum on the real line. A self-adjoint extension is fixed by
choosing the spectral parameter of the associated eigenvalue
equation equal to zero. The spectral problem is discussed in the context of
three different representations. For , the eigenvalue equation is
exactly solved in all these representations, in which square-integrable
solutions are explicity found. A set of constants of motion corresponding to
these quantum models is also obtained. Furthermore, the algebraic structure
underlying the quantum models is explored. This turns out to be a nonlinear
(quadratic) algebra, which could be applied for the determination of
approximate solutions to the eigenvalue equations.Comment: 24 pages, no figures, accepted for publication on JM
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
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