Exact solution for the nonlinear pendulum

This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is different from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75π (135◦) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ω in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135◦ the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics.This work was supported by the “Ministerio de Educación y Ciencia”, Spain, under project FIS2005-05881-C02-02, and by the “Generalitat Valenciana”, Spain, under project ACOMP/2007/020

Espécies do gênero Munida Leach (Crustacea, Decapoda, Galatheidae), distribuídas na costa do Brasil Species of the genus Munida Leach (Crustacea, Decapoda, Galatheidae) with distribution in the Brazilian coast

<abstract language="eng">The purpose of this work is to analyse the taxonomy and distributional patterns of the species of the genus Munida Leach, 1820 which occur along the Brazilian coast. The species of Munida present some peculiar characteristics, rarely found in any other marine group. They live in groups with great number of individuals (gregariousness) and with several species found together in the same population (sintopy). In addition, they exhibit a great degree of intraspeciftc variation and, paradoxally, a little variability among the species. These factors make a better understanding of the systematics of this group very difficult. In the present study the utilization of non variable characters was emphasized. As a corollary of this study, "species-complexes" were defined including groups of species morphologically very similar but yet well characterized. Sixteen species with distribution along the Brazilian coast were studied. An identification key is provided. A general discussion where the main problems on taxonomy and distributional patterns of species were analysed was done