# On the Stability of Circular Orbits

Bertrand's Theorem characterizes the force laws that govern stable circular orbits. It states that the only force laws permissible are the Hooke's Potential and Inverse Square Law. The proof of the theorem involves some perturbation techniques and series expansion.

The most natural things that comes to my mind when thinking about such a problem is that the effective force should be a restoring force for circular orbits to be stable.

$f_{\mathrm{eff}}(r) = \dfrac{l^2}{\mu r^3}-f(r) = 0$, for orbit to be circular.

$f'_{\mathrm{eff}}(r)<0$, for orbit to be stable. Assuming a power law, $f=Kr^n$, for the central force, solving it gives me the solution $n>-3$.

This is very weak compared to the statement of Bertrand's Theorem. Could someone explain to me the rationale behind perturbation technique used, and what is missing from my interpretation of 'stable' in my derivation?

• what do actually want? Want us to show you the derivation of Bertrand's Theorem? – user36790 Jun 20 '16 at 7:23
• I want to know why my proof is incorrect – Zhanfeng Lim Jun 20 '16 at 7:27