11 research outputs found
Autonomous three dimensional Newtonian systems which admit Lie and Noether point symmetries
We determine the autonomous three dimensional Newtonian systems which admit
Lie point symmetries and the three dimensional autonomous Newtonian Hamiltonian
systems, which admit Noether point symmetries. We apply the results in order to
determine the two dimensional Hamiltonian dynamical systems which move in a
space of constant non-vanishing curvature and are integrable via Noether point
symmetries. The derivation of the results is geometric and can be extended
naturally to higher dimensions.Comment: Accepted for publication in Journal of Physics A: Math. and Theor.,13
page
Generalizing the autonomous Kepler Ermakov system in a Riemannian space
We generalize the two dimensional autonomous Hamiltonian Kepler Ermakov
dynamical system to three dimensions using the sl(2,R) invariance of Noether
symmetries and determine all three dimensional autonomous Hamiltonian Kepler
Ermakov dynamical systems which are Liouville integrable via Noether
symmetries. Subsequently we generalize the autonomous Kepler Ermakov system in
a Riemannian space which admits a gradient homothetic vector by the
requirements (a) that it admits a first integral (the Riemannian Ermakov
invariant) and (b) it has sl(2,R) invariance. We consider both the
non-Hamiltonian and the Hamiltonian systems. In each case we compute the
Riemannian Ermakov invariant and the equations defining the dynamical system.
We apply the results in General Relativity and determine the autonomous
Hamiltonian Riemannian Kepler Ermakov system in the spatially flat Friedman
Robertson Walker spacetime. We consider a locally rotational symmetric (LRS)
spacetime of class A and discuss two cosmological models. The first
cosmological model consists of a scalar field with exponential potential and a
perfect fluid with a stiff equation of state. The second cosmological model is
the f(R) modified gravity model of {\Lambda}_{bc}CDM. It is shown that in both
applications the gravitational field equations reduce to those of the
generalized autonomous Riemannian Kepler Ermakov dynamical system which is
Liouville integrable via Noether integrals.Comment: Reference [25] update, 21 page
A transfer matrix method for the analysis of fractal quantum potentials
The scattering properties of quantum particles on fractal potentials at
different stages of fractal growth are obtained by means of the transfer matrix
method. This approach can be easily adopted for project assignments in
introductory quantum mechanics for undergraduates. The reflection coefficients
for both the fractal potential and the finite periodic potential are calculated
and compared. It is shown that the reflection coefficient for the fractal has a
self-similar structure associated with the fractal distribution of the
potential
Atomic time-of-arrival measurements with a laser of finite beam width
A natural approach to measure the time of arrival of an atom at a spatial
region is to illuminate this region with a laser and detect the first
fluorescence photons produced by the excitation of the atom and subsequent
decay. We investigate the actual physical content of such a measurement in
terms of atomic dynamical variables, taking into account the finite width of
the laser beam. Different operation regimes are identified, in particular the
ones in which the quantum current density may be obtained.Comment: 7 figure