Yukawa Force Orbit
A particle of mass m moves in a circle of radius R under the influence of a central attractive force
a) Determine the conditions on the constant α such that the circular motion will be stable.
b) Compute the frequency of small radial oscillations about this circular motion.
SOLUTION
The motion can be investigated in terms of the effective potential
(1)
where l is the angular momentum of the particle about the origin and
The conditions for a stable orbit are
(2)
where r = R is an equilibrium point for the particle in this now one dimensional problem. The requirement on the second derivative implies that the effective potential is a minimum, i.e., the orbit is stable to small perturbations. Substituting (1) into (S2), we obtain
(3)
The second condition of (2) gives
(4)
(3) gives
which, substituted into (4), yields
(5)
which implies that, for stability, a > R.
b) The equation for small radial oscillations with ζ = r - R is
(6)
The angular frequency for small oscillations given by (5) and (6) is found from
