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Consider a pendulum consisting of a compact mass on the end of light
inextensible string of length . Suppose that the mass is free to move
in any direction (as long as the string remains taut). Let the
fixed end of the string be located at the origin of our coordinate system.
We can define Cartesian coordinates, (, , ), such that
the axis points vertically upward. We can also define spherical
coordinates, (, , ), whose axis points along the axis. The latter coordinates are the most convenient, since is constrained to always take the value . However, the two angular coordinates,
and , are free to vary independently. Hence, this is
a two degree of freedom system.
The Cartesian coordinates can be written in terms of the angular coordinates
and . In fact,
Hence, the potential energy of the system is

(660) 
Also,

(661) 
Thus, the Lagrangian of the system is written

(662) 
Note that the Lagrangian is independent of the angular coordinate . It
follows that

(663) 
is a constant of the motion. Of course, is the angular momentum of
the system about the axis. This is conserved because neither the tension
in the string nor the force of gravity exert a torque about the axis.
Conservation of angular momentum about the axis implies that

(664) 
where is a constant.
The equation of motion of the system,

(665) 
yields

(666) 
or

(667) 
where use has been made of Equation (664).
Suppose that
. It follows that
. Hence, Equation (667) yields

(668) 
This, of course, is the equation of a simple pendulum whose motion is
restricted to the vertical plane see Section 3.10.
Suppose that
. It follows from Equation (664) that
: i.e., the pendulum bob rotates uniformly in a horizontal plane. According
to Equations (664) and (667),

(669) 
where
is the vertical distance of the plane of rotation below the pivot point. This type of pendulum is usually called a conical
pendulum, since the string attached to the pendulum bob sweeps out a
cone as the bob rotates.
Suppose, finally, that the motion is almost conical: i.e., the value
of remains close to the value . Let

(670) 
Taylor expanding Equation (667) to first order in , the zeroth
order terms cancel out, and we are left with

(671) 
Hence, solving the above equation, we obtain

(672) 
where

(673) 
Thus, the angle executes simple harmonic motion about its
mean value at the angular frequency .
Now the azimuthal angle, , increases by

(674) 
as the angle of inclination to the vertical, , goes between
successive maxima and minima. Suppose that is small.
In this case,
is slightly greater than . Now
if
were exactly then the pendulum bob would
trace out the outline of a slightly warped circle: i.e., something like the
outline of a potato chip or a saddle. The fact that
is slightly greater than means that this shape precesses about
the axis in the same direction as the direction rotation of the bob. The precession rate increases as
the angle of inclination increases. Suppose, now, that
is slightly less than . (Of course, can never exceed ).
In this case,
is slightly less than . Now
if
were exactly then the pendulum bob would
trace out the outline of a slightly tilted circle. The fact that
is slightly less than means that this shape
precesses about the axis in the opposite direction to the
direction of rotation of the bob. The precession rate increases as the
angle of inclination decreases below .
Next: Exercises
Up: Lagrangian Dynamics
Previous: Generalized Momenta
Richard Fitzpatrick
20110331