Energy and Angular Momentum Dependent Potentials with Closed Orbits

The Bertrand theorem concluded that; the Kepler potential, and the isotropic harmonic oscillator potential are the only systems under which all the orbits are closed. It was never stressed enough in the physical or mathematical literature that this is only true when the potentials are independent of the initial conditions of motion, which, as we know, determine the values of the constants of motion $E$ and $L$. In other words, the Bertrand theorem is correct only when $V\equiv V(r)\neq V(r,E,L)$. It has been derived in this work an alternative orbit equation, which is a substitution to the Newton's orbit equation. Through this equation, it was proved that there are infinitely many energy angular momentum dependent potentials $V(r,E,L)$ that lead to closed orbits. The study was done by generalizing the well known substitution $r=1/u$ in Newton's orbit equation to the substitution $r=1/s(u,E,L)$ in the equation of motion. The new derived equation obtains the same results that can be obtained from Bertrand theorem. The equation was used to study different orbits with different periodicity like second order linear differential equation periodicity orbits and Weierstrasse periodicity orbits, where interestingly it has been shown that the energy must be discrete so that the orbits can be closed. Furthermore, possible applications of the alternative orbit equation were discussed, like applications in Bohr-Sommerfeld quantization, and in stellar kinematics.