A simple method for obtaining both the hodograph and the orbit equation in the Kepler problem is presented. The method allows also for a very simple derivation of Hamilton's vector and shows at once that in velocity space all the regular Kepler orbits, i.e. those with a non-vanishing angular momentum, are circular. Resumen. Presentamos un método sencillo para obtener tanto la hodógrafa como la ecuación para las órbitas en el problema de Kepler. El método permite obtener en forma muy simple al vector de Hamilton y muestra de un vistazo que las órbitas del problema con momento angular diferente de cero son todas circulares en el espacio de las velocidades.
The hodograph, i.e. the path traced by a body in velocity space, was introduced by Hamilton in 1846 as an alternative method for studying certain dynamical problems. The hodograph of the Kepler problem was then investigated and shown to be a circle, it was next used to investigate some other properties of the motion. We here propose a new method for tracing the hodograph and the corresponding configuration space orbit in Kepler's problem starting from the initial conditions given and trying to use no more than the methods of synthetic geometry in a sort of Newtonian approach. All of our geometric constructions require straight edge and compass only. ResumenLa hodógrafa, i.e. la curva recorrida por un cuerpo en el espacio de las velocidades, fué propuesta por Hamilton en 1846 como una alternativa para investigar algunos problemas dinámicos. Se demostró entonces que la hodógrafa del problema de Kepler es una circunferencia y posteriormente se la usó para establecer algunas otras propiedades del movimiento. En este trabajo proponemos un método geométrico semi newtoniano para construir unaórbita elíptica partiendo de sus condiciones iniciales y de la correspondiente hodógrafa, empleando para ello métodos de la geometría sintética que requieren de la regla y del compásúnicamente.Classification Numbers: 03.20.+i, 95
The evolution and decay of a homogeneous flow over random topography in a rotating system is studied by means of numerical simulations and theoretical considerations. The analysis is based on a quasi-two-dimensional shallow-water approximation, in which the horizontal divergence is explicitly different from zero, and topographic variations are not restricted to be much smaller than the mean depth, as in quasi-geostrophic dynamics. The results are examined by comparing the evolution of a turbulent flow over different random bottom topographies characterized by a specific horizontal scale, or equivalently, a given mean slope. As in two-dimensional turbulence, the energy of the flow is transferred towards larger scales of motion; after some rotation periods, however, the process is halted as the flow pattern becomes aligned along the topographic contours with shallow water to the right. The quasi-steady state reached by the flow is characterized by a nearly linear relationship between potential vorticity and transport function in most parts of the domain, which is justified in terms of minimum-enstrophy arguments. It is found that global energy decays faster for topographies with shorter horizontal length scales due to more effective viscous dissipation. In addition, some comparisons between simulations based on the shallow-water and quasi-geostrophic formulations are carried out. The role of solid boundaries is also examined: it is shown that vorticity production at no-slip walls contributes for a slight disorganization of the flow.
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