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