For the spherical pendulum, with constant r = l, we thus have
Thus
is the conserved energy.
is conserved by Hamilton's equations, since
is conserved, we can view it as a constant parameter determining an effective potential for 
:
where
depends only on and the constant . Recalling that varies from 0 (upright) to 
(hanging vertically down), the two component
terms are drawn below.
When 
, the gravitational component is the only term and our
equilibrium point is at 
(hanging straight down). As
increases, the effective potential is a sum of both potential
terms. Since the term diverges at , the
effective potential minimum can no longer be there, for nonzero
. Instead, summing both contributions gives an effective
potential whose minimum is shifted lower: shifting
further and further to the left (lower values) for
greater and greater values of , until when the
component totally dominates, 
(that is, the
spherical pendulum rotates totally horizontally). Thus as
increases, the constant motions go from hanging down
stationary ( ), to rotating about the z-axis at fixed
, with the angle rising closer and closer to the
horizontal as the rotation frequency 
(hence )
increases. This effective potential and resulting motions are sketched
below.
corresponds to potential
(this is seen most easily by
applying 
in spherical coordinates, then using
our constraint.) Secondly, in cylindrical coordinates, infinitesimal
distances are
We thus have, applying our constraint 
,
This Lagrangian has partial derivatives
giving Euler-Lagrange equations
Thus
is the conserved energy.
The equations on the right just reproduce the relations between 
and 
, 
and 
. Using them to eliminate momenta and substitute into the Hamilton's equations on the left gives
reproducing the Euler-Lagrange equations.
implies that is constant;
increases at a constant rate (choosing 
). The E-L equation for z is a simple harmonic oscillator equation
with 
, with general solution 
. z thus oscillates about z=0 while
increases monotonically, as pictured. The motion is periodic
only if the particle returns to the same point after some basic
period; that is, returns to the same z and to 
. This
happens only if the period 
of z-motion and the period
of angular motion are rational multiples; otherwise the
particle never completes a z-cycle and a -cycle at the same
time.
Thus
which has asymptotes 
decreasing from infinity for small
r, and 
approaching zero from below as 
. In between the decreasing behavior at small r
and the increasing behavior at large r must be a turnover point
where 
has a minimum. The potential thus looks as sketched
above, with bound orbits for 
and scattering orbits for 
.
giving a minimum at
Thus
Thus
, giving
for positive 
. thus has a minimum at
Generally
which we simplify to
At 
we thus have
which
is always positive for 
, indicating stable circular
orbits (since deviations 
from have restoring force
).We thus have a frequency for radial oscillations
But by angular momentum conservation 
for a circular orbit at , so
Thus for our cases we have relations between the periods 
and
. This tells us how many cycles of the sinusoidal
variation in r we complete per 
-revolution in , as
drawn below.
