Saari’s Conjecture for the Collinear \u3cem\u3en\u3c/em\u3e-Body Problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian n-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations

Ressenyes

Index de les obres ressenyades: V. DÍAZ DE RADA IGÚZQUIZA, Pautas de consumo y ahorro en los albores del siglo XX

The Restricted 3-Body Problem on S1 : regularization and a particular solution

We study a special case of the two body problem when the particles are restricted to move in the space S1. We regularize all collisions using a symplectic transformation and classify the trajectories in four families. After that, we add a third infinitesimal body, getting four restricted three body problems on S1 corresponding to each one of the previous families. Then, we apply only one symplectic transformation that regularizes all the singularities due to binary collisions between the infinitesimal body with each primary. We show the global dynamics in one of the restricted problems, when the primaries are fixed at the poles of S1. We exhibit a particular set of solutions which takes place when the primaries perform hyperbolic motions only

Implementación de una formulación analítica del LEMP para estimar el desempeño de líneas de distribución frente a rayos

This paper presents the implementation of an analytical formulation to calculate the lightning electromagnetic pulse (LEMP) assuming a current wave-shape linearly rising with flat top and a transmission Line (TL) return-stroke model. It also describes the development of the expressions for the image dipoles required to calculate the vertical electric field, the azimuthal magnetic field and, specially, the horizontal electric field. The expressions to calculate the contribution of source dipoles were detailed in a previous publication by other authors. The complete formulation is used to calculate electromagnetic fields and lightning-induced voltages on a typical overhead distribution line. The results were compared with traditional formulas to calculate the LEMP (such as Rubinstein’s) and to calculate induced voltages (such as Rusck’s) showing errors below 1%. If a more complex wave shape was used (such as Heidler’s), errors below 5% were found. Additionally, the formula was employed to calculate the flashover rate of a distribution line above a ground with infinite and finite conductivity. Errors less than 5% were found compared to the results obtained in the IEEE 1410 Standard. On the other hand, the computation time required to the assessment of an overhead line indirect lightning performance is reduced by half when the analytical formula is used.Este artículo presenta la implementación de una formulación analítica para calcular el campo electromagnético producido por rayo (LEMP), asumiendo una forma de onda de la corriente tipo rampa plana y un modelo de línea de transmisión (TL) para la descarga de retorno. Se presenta el desarrollo de las expresiones para los dipolos imágenes necesarios para calcular el campo eléctrico vertical, el campo magnético azimutal y especialmente el campo eléctrico horizontal. Las expresiones para calcular la contribución de los dipolos fuentes se presentaron en una publicación previa de otros autores. La formulación completa se usó para calcular los campos electromagnéticos y las tensiones inducidas por rayo en una línea de distribución aérea típica. Los resultados fueron comparados con las fórmulas tradicionales para calcular el LEMP como la fórmula de Rubinstein y para calcular tensiones inducidas como la fórmula de Rusck mostrando errores menores al 1%. Si una forma de onda más compleja se usa como la fórmula de Heidler, se encuentran errores menores al 5%. Adicionalmente, la fórmula se empleó para calcular la tasa de fallas de una línea de distribución aérea para terrenos con conductividad finita. Se encontraron errores menores al 5% comparados con las obtenidas en el estándar IEEE 1410. De otro lado, el tiempo de cómputo requerido para la evaluación del desempeño ante rayos de líneas de distribución se reduce a la mitad cuando se usa la fórmula analítica