Before you starton this one,  it would be a very good idea to become familiar
with the Basic problem on the pendulum oscillator.

G.3.a.i)

Here's another look at everyone's favorite:

Thanks to Ben Halperin for the pendulum graphics.
In the grapics for this part, the sample value is used.

The position of the center of the bob at time is given by:

When , the pendulum looks like this:

This is the position of zero potential energy.
See the pendulum in a different position:

Look at:

For a given , the center of the bob is
         
units above the zeropotential energy line.
So, as the physics folks say:

The kinetic energy at the center of the bob at time is:

The total energy is:

One physical principle is that the the total energy of the undamped pendulum remains constant as time varies.
This is the same as saying that
         
for all times .
Execute this:

Compare your result with the undamped pendulum oscillator differential equation as propounded in the Basics:

Write up your conclusion.

G.3.a.ii)

Look at this:

The total energy at a time t at which y[t] = y and = x is:

Now look at this contour plot of energy[x,y];

On each plotted curve, energy[x,y] stays constant.
The lighter shading indicates larger values of energy[x,y].

Throw in a trajectory in the system by converting the undamped pendulum oscillator to a nonlinear system:

On each plotted curve, stays constant.
The lighter shading indicates larger values of .

Explain how you know that all the curves (not just the plotted trajectory) shown in the plot above are all trajectories in this system.

G.3.a.iii)

Take another look at the plot of some level curves of as above:

Discuss what other information you get from this plot.
What makes the pendulum operate with high energy?  Low energy?

There are some additional equilibrium points other than .
Where are they and what do they mean?
To get a handle on these questions, you might want to play with trajectories such as:

Or: