Day: January 24, 2017

Pendulum in imaginary time

Posted on by arkajad

Today’s post needs mathematical imagination and practice. Practice is important. For instance lot of practice is needed for a successful target slinging. Like in this video below

There are many stiles of slinging. For us probably the best is is Apache style, when the stone (representing our mathematical pendulum) is making vertical circles. We will practice the math behind such slings right below. But after we are done with that, we will check what happens with the sling when time is imaginary rather than than real. The problem of the pendulum with imaginary time has been posed by John Baez as an exercise in his notes on elliptic functions. It seems that the solution of this exercise has not been published yet.

But first let us considered the stone (or a bucket of water, if you wish) making vertical circles on a string. It is clear that there are certain cases (velocities) that are “forbidden”, for instance if you do not give the bucket on the string sufficient initial velocity, the water will fall on your head from the bucket.

The same with certain amusement park rides – not all speed are safe:

We have started looking into this problem yesterday in Cosmoplanetary pendulum, and Bjab, in his comments, proposed an answer, but without providing details. I promised to look into it and to post my own solution (in fact yesterday I have made an error in my calculations). Here is what I came with today. In fact, just a while ago, I have found this problem discussed, though not from the angle we need, in CBSE Class XI Supplementary Textual Material in POhysics, Unit IV, Motion in a Vertical Circle. I am borrowing a picture from this site:

It fits perfectly our needs, except that we use letter \mu for the mass. Letter m we use for the second argument of Jacobi elliptic functions discussed in previous posts. For a pendulum on a string to work the centrifugal force \mu\dot{\theta}^2 l experienced by the rotating mass must be greater or equal to the radial component of the gravitational force:

    \[\mu\dot{\theta}^2l\geq -\mu g \cos \theta.\]

The mass \mu cancels out, and g/l = \omega^2, thus

(1)   \begin{equation*}\dot{\theta}^2\geq -\omega^2 \cos \theta.\end{equation*}

For \dot{\theta} we use Eq. (1) from Cosmoplanetary pendulum, but without fixing the value of \omega:

(2)   \begin{equation*}\dot{\theta}^2=4\omega^2(\frac{1}{m}-\sin^2\frac{\theta}{2}).\end{equation*}

Using the last two equations we arrive at the inequality

    \[4(\frac{1}{m}-\sin^2\frac{\theta}{2})\geq-\cos \theta.\]

Now

    \[\cos \theta=1-2\sin^2\frac{\theta}{2},\]

therefore

    \[4(\frac{1}{m}-\sin^2\frac{\theta}{2})\geq 2\sin^2\frac{\theta}{2}-1\]

or

(3)   \begin{equation*}6 \sin^2\frac{\theta}{2}\leq \frac{4}{m}+1.\end{equation*}

Suppose first that m<1. Then \theta varies from 0 to 2\pi and the left hand side has the largest value for \sin^2\frac{\theta}{2}=1. Therefore

    \[\frac{4}{m}+1\geq 6,\]

or

(4)   \begin{equation*}m\leq\frac{4}{5}.\end{equation*}

We are done with m<1. When m>1, we already know from the discussion in previous posts that for the highest point of the pendulum we have \sin^2\frac{\theta}{2}=\frac{1}{m}.
Therefore

    \[ \frac{6}{m}\leq \frac{4}{m}+1,\]

or

(5)   \begin{equation*}m\geq 2.\end{equation*}

It follows that the forbidden region is for m between 4/5 and 2 – as correctly predicted by Bjab in his comment yesterday.

We now come to the second issue – that of imaginary time. Here is the exercise form notes by John Baez: The Pendulum, Elliptic Functions and Imaginary Time. There we find the following “exercise”:

11. Show that making the replacement

    \[t\mapsto it\]

in Newton’s law is equivalent to reversing the sign of all forces.

In the present problem, this amounts to reversing the force of gravity, making it pull the pendulum up. But an upside-down pendulum is just another pendulum. Therefore the function \mathrm{sn}(it, k) must also be periodic as a function of t. This suggests that \mathrm{sn}(z, k), as a function of z\in\mathbb{C}, is periodic in both the real and imaginary directions. And it’s true!

So, the pendulum gives a physical explanation of the fact that elliptic functions are periodic in two directions on the complex plane!

12. Prove, as rigorously as you can, that \mathtm{sn}(z, k) is periodic in two directions. You can do this either by fleshing out the above argument, or by studying the integral in equation (3) and worrying about those branch points. In fact we have

    \[\mathrm{sn}(z + 4K, k) = \mathrm{sn}(z, k),\quad \mathrm{sn}(z + 2iK',k) = \mathrm{sn}(z, k)\]

where for0 < k < 1

    \[K=\int_0^1\frac{dy}{\sqrt{(1-y^2)(1-k^2y^2}}\]

and

    \[K'=\int_0^{1/k}\frac{dy}{\sqrt{(y^2-1)(1-k^2y^2}}.\]

It seems that the first nine exercises from these notes (for MATH 241 course) has been solved by Toby Barlets and posted. But not 11 and 12. Can we make sense of Baez’idea of “upside-down pendulum is just another pendulum” and use it to derive the formula with 2K’ imaginary period?

Who can do it?