From circular paths to elliptic orbits: a geometric approach to Kepler's motion

Abstract

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

Tracing a planet’s orbit with a straight edge and a compass with the help of the hodograph and the Hamilton vector

“…25 Our method is an attempt to exhibit in simple terms the geometric beauty of dynamics -beauty that captured the heart of Newton himself. 3,5,14,22 Our method can be profitably applied to other related problems. For example, what if we wish to describe the trajectory of a comet?…”

“…1 Although this way of solving the problem apparently involves a detour, it ends up being one of the simplest ways of finding the orbit. 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 . Moreover, this approach makes it straightforward to obtain an additional constant of the motion, namely the Hamilton vector.…”

A new look at the Feynman ‘hodograph’ approach to the Kepler first law

“…Such a derivation starts by unveiling (rediscovering) a curious property: the Kepler hodographs have an exact circular character, but this circle is not centred in the origin of velocity space (see e.g. [17][18][19][20][21][22][23]).…”

Quantum hologram and relativistic hodogram: Magnetic resonance tomography and gravitational wavelet detection

“…In any case, the angle between two radial velocity vectors, as measured around the velocity director circle, is equal to the phase angle between corresponding position vectors. These important facts on the projective geometry of the Kepler problem have been discovered by Sir William Rowan Hamilton (1805-1865) in 1846 when he studied an alternative for solving dynamical problems [29][30][31]44]. They are straightforward within the coadjoint orbit picture of the unitary dual G of the Heisenberg nilpotent Lie group G and permit an inclusion of special relativity into geometric quantization.…”