We describe a geometrical method for tracing a planet's orbit using its velocity hodograph, that is, the path of the planet's velocity. The method requires only a straight edge, a compass, and the help of the hodograph. We also obtain analitically the hodograph and some of the features of the motion that can be obtained from it.
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 classical motion of a particle in a 3D inverse square potential with negative energy, E, is shown to be geodesic, i.e., equivalent to the particle's free motion on a non-compact phase space manifold irrespective of the sign of the coupling constant. We thus establish that all its classical orbits with E < 0 are unbounded. To analyse the corresponding quantum problem, the Schrödinger equation is solved in momentum space. No discrete energy levels exist in the unrenormalized case and the system shows a complete “fall-to-the-center” with an energy spectrum unbounded by below. Such behavior corresponds to the non-existence of bound classical orbits. The symmetry of the problem is SO(3) × SO(2, 1) corroborating previously obtained results.
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