51 research outputs found
Complete sets of invariants for dynamical systems that admit a separation of variables
Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} = ij. The 2n–1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion
Hopf instantons in Chern-Simons theory
We study an Abelian Chern-Simons and Fermion system in three dimensions. In
the presence of a fixed prescribed background magnetic field we find an
infinite number of fully three-dimensional solutions. These solutions are
related to Hopf maps and are, therefore, labelled by the Hopf index. Further we
discuss the interpretation of the background field.Comment: one minor error corrected, discussion of gauge fixing added, some
references adde
On algebraic construction of certain integrable and super-integrable systems
We propose a new construction of two-dimensional natural bi-Hamiltonian
systems associated with a very simple Lie algebra. The presented construction
allows us to distinguish three families of super-integrable monomial potentials
for which one additional first integral is quadratic, and the second one can be
of arbitrarily high degree with respect to the momenta. Many integrable systems
with additional integrals of degree greater than two in momenta are given.
Moreover, an example of a super-integrable system with first integrals of
degree two, four and six in the momenta is found.Comment: 37 page
Fermion Zero Modes in Odd Dimensions
We study the zero modes of the Abelian Dirac operator in any odd dimension.
We use the stereographic projection between a dimensional space and a
sphere embedded in a dimensional space. It is shown that the
Dirac operator with a gauge field of uniform field strengths in has
symmetries of SU()U(1) which is a subgroup of SO(). Using group
representation theory, we obtain the number of fermion zero modes, as well as
their explicit forms, in a simple way.Comment: 14 page
Multiple zero modes of the Dirac operator in three dimensions
One of the key properties of Dirac operators is the possibility of a
degeneracy of zero modes. For the Abelian Dirac operator in three dimensions
the construction of multiple zero modes has been sucessfully carried out only
very recently. Here we generalise these results by discussing a much wider
class of Dirac operators together with their zero modes. Further we show that
those Dirac operators that do admit zero modes may be related to Hopf maps,
where the Hopf index is related to the number of zero modes in a simple way.Comment: Latex file, 20 pages, no figure
Review on possible gravitational anomalies
This is an updated introductory review of 2 possible gravitational anomalies
that has attracted part of the Scientific community: the Allais effect that
occur during solar eclipses, and the Pioneer 10 spacecraft anomaly,
experimented also by Pioneer 11 and Ulysses spacecrafts. It seems that, to
date, no satisfactory conventional explanation exist to these phenomena, and
this suggests that possible new physics will be needed to account for them. The
main purpose of this review is to announce 3 other new measurements that will
be carried on during the 2005 solar eclipses in Panama and Colombia (Apr. 8)
and in Portugal (Oct.15).Comment: Published in 'Journal of Physics: Conferences Series of the American
Institute of Physics'. Contribution for the VI Mexican School on Gravitation
and Mathematical Physics "Approaches to Quantum Gravity" (Playa del Carmen,
Quintana Roo, Mexico, Nov. 21-27, 2004). Updates to this information will be
posted in http://www.lsc-group.phys.uwm.edu/~xavier.amador/anomalies.htm
The last integrable case of kozlov-Treshchev Birkhoff integrable potentials
We establish the integrability of the last open case in the Kozlov-Treshchev
classification of Birkhoff integrable Hamiltonian systems. The technique used
is a modification of the so called quadratic Lax pair for Toda lattice
combined with a method used by M. Ranada in proving the integrability of the
Sklyanin case.Comment: 13 page
Spinor Field in Bianchi type-I Universe: regular solutions
Self-consistent solutions to the nonlinear spinor field equations in General
Relativity has been studied for the case of Bianchi type-I (B-I) space-time. It
has been shown that, for some special type of nonliearity the model provides
regular solution, but this singularity-free solutions are attained at the cost
of broken dominant energy condition in Hawking-Penrose theorem. It has also
been shown that the introduction of -term in the Lagrangian generates
oscillations of the B-I model, which is not the case in absence of
term. Moreover, for the linear spinor field, the term provides
oscillatory solutions, those are regular everywhere, without violating dominant
energy condition.
Key words: Nonlinear spinor field (NLSF), Bianch type -I model (B-I),
term
PACS 98.80.C CosmologyComment: RevTex, 21 page
Brane world corrections to Newton's law
We discuss possible variations of the effective gravitational constant with
length scale, predicted by most of alternative theories of gravity and unified
models of physical interactions. After a brief general exposition, we review in
more detail the predicted corrections to Newton's law of gravity in diverse
brane world models. We consider various configurations in 5 dimensions (flat,
de Sitter and AdS branes in Einstein and Einstein-Gauss-Bonnet theories, with
and without induced gravity and possible incomplete graviton localization), 5D
multi-brane systems and some models in higher dimensions. A common feature of
all models considered is the existence of corrections to Newton's law at small
radii comparable with the bulk characteristic length: at such radii, gravity on
the brane becomes effectively multidimensional. Many models contain superlight
perturbation modes, which modify gravity at large scale and may be important
for astrophysics and cosmology.Comment: Brief review, 16 pages, 92 references. Some description and
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