Free fall on a fixed center and harmonic oscillator.
Newton’s equation for the free fall (Unnecessary constants have been removed) :
That equation is best integrated analytically by considering the energy first integral, (
Here is the solution in implicit form t=time(x) :
A forward numerical integration suspects the presence of a singularity near t=1.11... :
Changing both dependant and independant variables in Mathematica :
1) The change of Sundman (x=
and dt=
dT) :
Combining the two former equations leads to a harmonic oscillation of X as a function of T.
2) No monomial point transformation, x=
and t=
(where a, b, c, d are constants whatever their numerical values) can achieve such a reduction :
