This  notebook  discusses methods of solving the simple pendulum
equation.  It  is  the first in a series of notebooks showing how
Mathematica can be used to find exact power series solutions (to
arbitrary order) of several differential equations.

A point mass at the end of a massless rigid rod is constrained to move
without friction on a vertical circle about a fixed point.  If the
polar angle (phi) is measured counter-clockwise from vertically
downward, the dimensionless form of the simple pendulum equation for
the polar angle  (phi) is:

The authors introduced a method of using Mathematica's symbolic
programming to find exact power series solutions of differential
equations (exact, to arbitrary order) [Power series approximation to
solutions of nonlinear systems of differential equations, Pickett et
al, Am. J. Phys. 61 (1), January 1993].  The method was extended to
nonlinear partial differential equations [Power series solution of
linear and nonlinear partial differential equations via symbolic
programming, Pickett et al, Computers in Physics, Vol 7, No 1, Jan/Feb
1993].

If  the order of the power series is kept low, even floating point
arithmetic often works fairly well.  However, when more accuracy (and
thus a higher order) is required, floating point errors ultimately
cause a dramatic decrease in accuracy.

The authors have applied the method to many of the most important
linear and nonlinear ordinary and partial differential equations of
classical physics with excellent results.  Applications to quantum
mechanics may be even more rewarding.