Jacobi amplitude function appeared yesterday in the episode Derivatives of Jacobi elliptic am, sn, cn, dn. We have derived a beautiful simple differential equation satisfied by this beautiful function
Today we will see that, after rescaling, this is a perfect fabric for making the nonlinear outfit for the mathematical pendulum.
Wikipedia has smart animations showing pendulum’s motion for different kinetic energies, for instance
Initial angle of 45°
Pendulum with enough energy for a full swing.
It has also another little animation showing the angle avrying with time.
But this last picture is not well adapted for a pendulum that is making full swings around the circle. Therefore I will refer to the picture that I was already using in The case of the swinging pendulum:
This last picture has many more features depicted than I will need. I will need only the angles and and the length of the pendulum . The mass of the swinging point P I will denote by . Usually it is denoted by but we will use for the square of the modulus of the Jacobi amplitude function
For solving our pendulum problem we will only need conservation of energy.
When the pendulum swings, the angle changes with time We will use the dot to denote time derivative of
The linear velocity of the pendulum is , therefore the kinetic energy is
For the potential energy we will choose the zero of the potential at the bottom, Denoting by the height of the mass with respect to the lowest level, we have
For we have for we have Here it is useful to introduce the half-angle We know from trigonometry that
Potential energy is that is
Maximum kinetic energy is at the bottom, for . At this point we have also minimum potential energy, since at the point. Maximal potential energy is at the top:
The character of the motion will depend on the ratio When this ratio is there will be not enough kinetic energy to rise the swinging mass to the top, and the pendulum will oscillate back and forth. But when the ratio then
even at the top the mass will have a nonzero speed, and the pendulum will be making full circles. We denote this important ratio by
We now write the conservation of energy equation:
We also introduce defined as
This is the expression for the standard angular frequency for a linear pendulum, fo small oscillations.
With all these substitutions and simplifications Eq. (5) can be written in the following form:
The last equation is almost identical with the equation (1) satisfied by the amplitude function, except for the coefficient in front of on the left. But this can be easily accomodated by changing the time scale. With we notice that
Comparing with Eq. (7) we see that the solution of the pendulum equation is
In the next post we will look closer at this solution and try to understand its meaning.