‘Aequat causa effectum’ (the cause is equal to the effect) is an old
metaphysical rule, which seems, however, to be violated for the single trajectories in
spherically symmetric force fields, as the Kepler orbits are not spherically symmetric.
The reason for the latter is the lower symmetry of the initial conditions. The rule is
reestablished, if appropriate sets of trajectories (i.e, sets of initial conditions) are
considered. The corresponding hodographs are also explored. The fact, that the Kepler
orbits are plane, is derived by means of logical and symmetry arguments, without
using angular momentum conservation. Spherically symmetric sets of Bohr orbitals
are constructed by means of a sum rule for the spherically harmonic functions
(Unsöld’s theorem). This results in a less known similarity between classical and
quantum mechanical systems. Metaphysical rules do not solve concrete physical
problems, but encourage to explore them more profoundly.
Keywords: Aequat causa effectum, Hodograph, Initial conditions, Kepler orbits,
Metaphysics, Spherically symmetric force fields, Unsöld’s theorem.
