332 research outputs found
Jacobi equations using a variational principle
A variational principle is proposed for obtaining the Jacobi equations in
systems admitting a Lagrangian description. The variational principle gives
simultaneously the Lagrange equations of motion and the Jacobi variational
equations for the system. The approach can be of help in finding constants of
motion in the Jacobi equations as well as in analysing the stability of the
systems and can be related to the vertical extension of the Lagrangian
formalism. To exemplify two of such aspects, we uncover a constant of motion in
the Jacobi equations of autonomous systems and we recover the well-known
sufficient conditions of stability of two dimensional orbits in classical
mechanics.Comment: 7 pages, no figure
Non-unitary representations of the SU(2) algebra in the Dirac equation with a Coulomb potential
A novel realization of the classical SU(2) algebra is introduced for the
Dirac relativistic hydrogen atom defining a set of operators that, besides,
allow the factorization of the problem. An extra phase is needed as a new
variable in order to define the algebra. We take advantage of the operators to
solve the Dirac equation using algebraic methods. To acomplish this, a similar
path to the one used in the angular momentum case is employed; hence, the
radial eigenfuntions calculated comprise non unitary representations of the
algebra. One of the interesting properties of such non unitary representations
is that they are not labeled by integer nor by half-integer numbers as happens
in the usual angular momentum representation.Comment: 20 pages 1 eps figure in a single zipped file, submitted to J. Math.
Phy
Bioindicators as a tool for monitoring and control biofilm reactors : a simplified approach
The control of wastewater treatment plants requires an extensive monitoring programme based
on physicochemical costly routine analysis. Microbial populations are well known indicators of
operational conditions in biological reactors. This research was carried out by the CENTA and the
University of Minho in the scope of the Erasmus programme. The aim of the work was the
development of a simplified approach, based on biological indicators, for monitoring small biofilm
wastewater treatment processes. In that regard, protozoa and metazoan were monitored at
CENTA experimental plants - namely a trickling filter and a rotating biological contactor -, and
correlated with influent wastewater and effluent composition. The relationship between
wastewater performance and the microbiological composition of the biofilm was identified and
assessed. Results indicate that a methodology based on the bioindicators provides useful data
for process monitoring and control of small wastewater treatment plants, thus diminishing the
associated costs of routine analysis and providing information when such analysis aren’t easily
available
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