Day: January 11, 2017

Jacobi elliptic functions – Definition as trigonometry

Posted on by arkajad

Jacobi elliptic functions are needed for writing down explicit solutions of the free asymmetric top, and I decided to define them in a way that somehow resembles the way in which ordinary sinus and cosinus functions are defined in trigonometry. In my Polish blog (Sinus zdeformowany, Cosinus zdeformowany, Zdeformowna jedynka)I used for this purpose the method proposed by William A. Schwalm, that I have mentioned in my previous post on Jacobi.

Yesterday professor Schwalm kindly informed me about his debate with one of the Wikipedia librarians, one responsible for the “Definition as trigonometry” section of the Wikipedia entry on Jacobi elliptic functions. After exchanging several emails with prof. Schwalm I decided, if only just for change, to present here the method used in Wikipedia, a variation of the method used by W. A. Schwalm in his handout notes “Elliptic Functions sn, cn, dn, as Trigonometry“ and in his book “Lectures on Selected Topics in Mathematical Physics: Elliptic Functions and Elliptic Integrals“.

I must, however, say right away that I am not happy with what I have found in Wikipedia. I think that what is in Wikipedia is, at least, very misleading. And I will explain toward the end of this post why I think it is so.

Let’s go to work. We start with drawing an ellipse

(1)   \begin{equation*}\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1.\end{equation*}

and we choose one with a=1 and b\geq 1.

Wikipedia for some reason avoids showing any such drawing, so what I am presenting here is a drawing much like in Schwalm’s “Elliptic Functions sn, cn, dn, as Trigonometry“, but adapted to Wikipedia article conventions.


Define

(2)   \begin{equation*}m=1-\frac{1}{b^2}.\end{equation*}

Consider a point Q(x,y) on the ellipse, with

(3)   \begin{equation*} x=r \cos \theta,\end{equation*}

(4)   \begin{equation*} y= r \sin \theta.\end{equation*}

Using the equation of the ellipse we can eliminate x,y and get for r

(5)   \begin{equation*} r=r(\theta,m)=\frac{1}{\sqrt{1-m\,\sin^2 \theta}}.\end{equation*}

Here is the calculation
We have x^2+y^2=r^2 and x^2+y^2/b^2=1. Therefore x^2=r^2-y^2 and so r^2-y^2+y^2/b^2=1 or y^2(1/b^2-1)=1-r^2. Thus y^2=(r^2-1)/m. But y^2=r^2\sin^2\theta. Therefore (r^2-1)/m=r^2\sin^2\theta, and we can calculate r^2=1/(1-m\sin^2\theta).
We define now

(6)   \begin{equation*}u=u(\phi,m)=\int_0^\phi r(\theta,m)\,d\theta.\end{equation*}

Then y'=\sin \phi is defined to be Jacobi function \mathrm{sn}(u).
In other words: if

(7)   \begin{equation*} u=\int_0^\phi \frac{1}{\sqrt{1-m\,\sin^2 \theta}}\,d\theta, \end{equation*}

then we define

(8)   \begin{equation*}\mathrm{sn}(u,m)=\sin \phi.\end{equation*}

Notice that we are in agreement with the definition that can be found, for instance, in “Handbook of Mathematical Functions“, Ed. Milton Abramowitz and Irene A. Stegun:

We are also in agreement with the definition of Jacobi Elliptic Functions in Wolfram MathWorld.

The advantage of using the picture above is that the integrand in Eq. (7) has a simple interpretation as r(\theta,m) of Eq. (5).

Thus so far so good. But the Wikipedia tries to do even better, and that is where it gets worse. Here is the bad part, extract from Wikipedia:

First they introduce undefined “angular component of the arc length” and “angular arc length“. There is no such thing as “angular component of the arc length“, it does not make any sense. In mathematics we should avoid using undefined terms. Introducing such term is a crime – unless the meaning is obvious. But here the meaning is not obvious at all. The fact that “angular arc length” is in quotation marks is not an excuse. It is, in my opinion, misleading. Then there is a mysterious sentence about “angular arc length” and “total arc length of hyperbolas“. Then the formula (6) is interpreted as “angular arc length for the ellipse“. That is very misleading. The arc length for ellipse, like for any other curve, is defined as the integral

(9)   \begin{equation*} \int_0^\phi\sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}\,d\theta.\end{equation*}

For our ellipse it can be transformed into

(10)   \begin{equation*}\frac{1}{b}\int_0^\phi r^2\sqrt{1+b^2-r^2}d\theta,\end{equation*}

which is certainly not the same as \int_0^\phi rd\theta as suggested by Wikipedia.

Calculation:
We have

    \[ x=r\cos\theta,\, y= r\sin\theta,\,r=\frac{1}{\sqrt{1-m\,\sin^2 \theta}}. \]

    \[\left(\frac{dx}{d\theta}\right)^2=\frac{(1-m)^2\sin^2\theta}{(1-m\sin^2\theta)^3},\]

    \[\left(\frac{dy}{d\theta}\right)^2=\frac{\cos^2\theta}{(1-m\sin^2\theta)^3}==\frac{1-\sin^2\theta}{(1-m\sin^2\theta)^3}.\]

For \sin^2\theta we substitute now \frac{1-1/r^2}{m} to get

    \[\left(\frac{dx}{d\theta}\right)^2=\frac{1}{m}(1-m)^2r^4(r^2-1),\]

    \[\left(\frac{dy}{d\theta}\right)^2=\frac{1}{m}(r^4+(m-1)r^6).\]

Adding together:

    \[\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2=r^4(2-m+r^2(m-1))=\frac{r^4}{b^2}(1+b^2-r^2).\]

In order to check Eq. (10) and get the circumference p of the ellipse (for b=2), I have numerically calculated the integral (10) from 0 to \pi/2 and multiplied by 4 to get p=9.68845. On the other hand Ramanujan suggested a simple approximate formula for the perimeter of an ellipse

(11)   \begin{equation*} p\approx \pi\left[3(a+b)-\sqrt{(3a+b)(3b+a)} \right].\end{equation*}

Using Ramanujan formula I get essentially the same answer p=9.68842. But if I would use “angular arc length for the ellipse” of Wikipedia, I would get 8.62606 as the answer – evidently wrong.

Of course there is yet another way of calculating the perimeter of the ellipse. We can parametrize ellipse as

    \[ x=a\cos\tau,\]

    \[y= b\sin \tau.\]

The parameter \tau is then not the same as the angle \theta above. But now for a=1, b=2, for the perimeter we easily get

(12)   \begin{equation*} p=4\int_0^{\pi/2} \sqrt{\left(\frac{dx}{d\tau}\right)^2+\left(\frac{dy}{d\tau}\right)^2}d\tau=8\int_0^{\pi/2} \sqrt{1-m\,\sin^2 \tau}\,d\tau.\end{equation*}

It is good to know, but it is irrelevant for our purpose.