Motion of a planet around a fixed center : compressions of angular data.
1 Newton’s compression.
Newton’s equations (here in hamiltonian form) - together with the initial conditions set - accurately compress the data of measured angles θ (I1, I2 and I3 are the three autonomous invariants) :
Initial conditions :
Autonomous invariants :
Period of the motion :
Trajectory :
Distance between center and focus :
2 Hipparchus-like compression :
A pure kinematical compression of θ as a function of t is possible on the ground of the Fourier-like false position, {x(t), y(t)} = {
+ ... } :
First approximation : one harmonic with period T (a1 and b1) :
A very off-centered elliptic trajectory :
A geometrical construction involving two generating circles :
How well θ(t) is approximated (Newton in black and Hipparchus in red) :
Absolute error :
Second approximation : two harmonics with periods T and T/2 (a1, a2, b1 and b2) :
This time, θ(t) is well approximated (Newton in black and Hipparchus in red) :
The trajectory is no more an ellipse (!) :
A geometrical construction involving four generating circles :
