Stability of a circular orbit in a central force field

There are at least two approaches to the stable circular orbits for the two-body problem in a central potential U(r).

• Lagrange's equations
• Effective potential energy

Lagrange's equations. Lagrangian L = T-U

• L = 1/2 m (r-dot^2 + r^2 theta-dot^2) - U(r)
• m = reduced mass = m1 m2/(m1+m2)
• equations of motion
• m r-dot-dot = m r theta-dot^2 -U'(r)
• d/dt(mr^2 theta-dot) = 0 => mr^2 theta-dot = l = angular momentum
• substitute in the mr-dot-dot equation for theta-dot and get
• m r-dot-dot = l^2/mr^3 -U'(r)
• For a circular orbit, we need r-dot-dot = 0 at r=ro
• this requires mr theta-dot^2 = U'(ro), or that (substituting for theta-dot)
• l^2/(mro^3) = U'(ro)
• For a stable circular orbit, we need small oscillations to occur around ro
• let r=ro + x, where x<<ro
• then the equation of motion becomes
• m x-dot-dot = l^2/(m (ro+x)^3) -U'(ro+x)
• we binomially expand the first term on the rhs and taylor-expand the 2nd.
• this gives m x-dot-dot = l^2/(mro^3)(1-3 x/ro) -U'(ro) - x U''(ro).
• cancelling the leading terms (due to circular orbit condition) gives
• m x-dot-dot = - x [3l^2/(mro^4) + U''(ro)] .
• this will oscillate if the expression in square brackets is greater than zero.
• substituting from the circular orbit conditon gives
  3/r U'(r) + U''(r) >0 for a stable circular orbit

Effective potential energy, Ueff(r)

• E = 1/2 m (r-dot^2 + r^2 theta-dot^2) +U(r)
• substitute for theta-dot from angular momentum in energy equation and get
• E = m/2 r-dot^2 + l^2/(2mr^2) + U(r)
• Since the last two terms are functions of r, we can regard them as an 'effective potential'
• Ueff = l^2/(2mr^2) + U(r)
• For a circular orbit we need Ueff'(r) = 0 at r=ro
• This requires -l^2/(mr^3) + U'(r) = 0
• For a stable circular orbit we need Ueff''(r) >0 at r=ro
• this requires 3l^2/(mr^4) + U''(r) >0 at r=ro
• substituting in from the circular orbit condition we get
  3/r U'(r) + U''(r) >0 for a stable orbit

These two approaches give the same conditions for a stable circular orbit.

• The effective potential approach fits right into what we did before for frequency of small oscillations near a potential minimum
• The expansion r = ro+x is good in that it reminds us of the method of series expansions to solve problems.