Consider a pendulum created by connecting a large mass to a thin stiff
rod, suspended from a pivot point in which it is able to move
about a full 360 degree circle. Unless the pendulum is pointed
directly downward, or balanced perfectly above the pivot point,
the gravitational force will result in a torque which pulls the
pendulum downwards.
This results in a angular acceleration for the pendulum of
For small angles , the value of is approximately
, and with this approximation the equation of motion is given by
The equation whose second derivative is a constant multiplied by the
opposite of itself is well known, and the solution is readily given.
However, this solution was based upon the assumption that the
initial displacement was small.
While the original problem may not have a closed form solution,
a numerical integration can yield interesting results as to the
motion of pendulums at large displacements.
In order to numerically solve the equation of motion without the
small angle approximation, we first take our equation which is
written in terms of a second derivative, and rewrite it in terms
of two equations written in terms of a first derivative. We do this
by realizing that the derivative of the angle of the pendulum is
just the angular velocity, .
and we can write the second derivative as the derivative of the first derivative
giving us as our two equations written in terms of first derivatives,
and
These two equations make up a system of ordinary differential
equations, and these equations, together with the initial position
and velocity of the pendulum, make up what is referred to as an
initial value problem.