Comments on the dynamical invariants of the Kepler and harmonic motions

Abstract

The properties of the Kepler motion can be derived using the invariant Hamilton vector as well as the invariant Laplace vector. No simple invariant tensor can be constructed from these vectors. The corresponding problem for the harmonic oscillator is considered. Although Laplace and Hamilton vectors can be defined, using the invariant Laplace tensor is much simpler.

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

“…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.…”

On Peres approach to Fradkin-Bacry-Ruegg-Souriau’s perihelion vector

“…As established by Jauch and Hill [9] in the 2D case and by Fradkin [10] in the 3D case, the isotropic harmonic oscillator shares the same feature and admits also a supplementary dynamical conserved quantity but of tensorial type, the Jauch-Hill-Fradkin tensor [1,6,11].…”

Hamilton symmetry in relativistic Coulomb systems

“…The Hamilton symmetry is therefore an extension of central symmetry. Further information and discussions regarding the classical KC analytic hodograph solution may be found in various publications [4,5,6,7,8,9,10,11].…”

“…The hodograph method -studying the dynamics of a system in velocity space -was originally invented by Hamilton [1] and successfully applied [2,3] to prove geometrically the relation between Kepler's laws and Newton's law of universal attraction. Although largely unfamiliar with, it is a method much simpler and more elegant to study Newtonian Kepler/Coulomb (KC) systems than the familiar analytic solutions in ordinary space [4,5,6,7,8,9,10,11].…”