I am solving dynamics and unable to digest author's views regarding apse. Help from the community is needed.
Theorem :- If the central acceleration p is a single valued function of the distance,
every apse line divides the orbit into equal and symmetrical portions
Author's goes on to prove the below result
v2 = C - ∫ Pdr
{v - velocity , P - central acceleration , r - radial distance of particle assumed to be moving under the effect of central acceleration P , pole of the acceleration lies on origin}
Author states that since acceleration and velocity are single valued for r , they are same for same value of r.Hence the direction of motion is independent[understood and agreed].
So if the motion of the particle is reversed at a particular apse , it will trace a symmetrical orbital path on the other side of apsidal distance.[not understood]
I accept that the particle upon changing the direction of motion will trace the same path from where it has arrived , it nowhere leads to the central orbit being symmetrical about the apse.