Jacobi elliptic functions – Definition as trigonometry

[latexpage]
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 \begin{equation}\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1.\label{eq:el}\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
\begin{equation}m=1-\frac{1}{b^2}.\end{equation}
Consider a point $Q(x,y)$ on the ellipse, with
\begin{equation} x=r \cos \theta,\end{equation}
\begin{equation} y= r \sin \theta.\end{equation}
Using the equation of the ellipse we can eliminate $x,y$ and get for $r$
\begin{equation} r=r(\theta,m)=\frac{1}{\sqrt{1-m\,\sin^2 \theta}}.\label{eq:r}\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
\begin{equation}u=u(\phi,m)=\int_0^\phi r(\theta,m)\,d\theta.\label{eq:u}\end{equation}
Then $y’=\sin \phi$ is defined to be Jacobi function $\mathrm{sn}(u).$
In other words: if \begin{equation} u=\int_0^\phi \frac{1}{\sqrt{1-m\,\sin^2 \theta}}\,d\theta, \label{eq:udef}\end{equation}
then we define \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. (\ref{eq:udef}) has a simple interpretation as $r(\theta,m)$ of Eq. (\ref{eq:r}).

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 undefinedangular 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 (\ref{eq:u}) 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
\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
\begin{equation}\frac{1}{b}\int_0^\phi r^2\sqrt{1+b^2-r^2}d\theta,\label{eq:int}\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. (\ref{eq:int}) and get the circumference $p$ of the ellipse (for $b=2$), I have numerically calculated the integral (\ref{eq:int}) 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
\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 ellipseof 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
\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.

10 thoughts on “Jacobi elliptic functions – Definition as trigonometry

  1. “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“”

    “angular component of the arc length” is defined as rdtheta
    “angular arc length” is defined as integral(rdtheta)
    One cannot argue with definitions.

    1. [latexpage]
      “One cannot argue with definitions.”

      I disagree. One can and sometimes one should argue with definitions. Especially when they are not explicitly stated as definitions.

      But suppose the author in Wikipedia would state: I define “angular component of the arc length” as $r\, d\theta$
      I define “angular arc length” is defined as $\int r\,d\theta$.

      Even then I would argue that these are bad and misleading definition.
      Suppose I write “I define 3 as arc component of 5.$ The reader would have legitimate rights to ask why are you choosing these terms? It is silly and it is misleading.

      Consider the special right triangle.

      I can define 4 as “horizontal component of 5” and define 3 as its vertical component. But that would be misleading. Arc length can not be decomposed into a sum of its angular and radial components much like 5 is not the sum of 3 and 4.

      The author of the Wikipedia entry should state that it is his own, personal, invention and that it has nothing to do with the true arc length whatsoever. Wikipediashould not the place where people would post their own poetry pieces.

      1. “I can define 4 as “horizontal component of 5” and define 3 as its vertical component. But that would be misleading. Arc length can not be decomposed into a sum of its angular and radial components much like 5 is not the sum of 3 and 4.”

        Are “component” and “addend” synonyms?

        1. “Are “component” and “addend” synonyms?”

          One should try to be clear, think about potential readers, and how they can interpret it. Advertising agencies are making use of the rules how our minds tend to associate certain words with certain concepts. But when teaching mathematics potentially dangerous associations should be avoided.

          “Component” is usually associated with “part of”. Here it is potentially dangerous and misleading.

          It is true that a vector has “components”, but length is a scalar, not a vector. For scalars “component” has a different meaning.

  2. “But if I would use “angular arc length for the ellipse” of Wikipedia, I would get 8.62606 as the answer – evidently wrong.”

    I think, Ark, that there are some relations between “angular arc length” and “arc length” – e.g. the first is not greater then the second.

    1. “I think, Ark, that there are some relations between “angular arc length” and “arc length” – e.g. the first is not greater then the second.”

      Sure. Nevertheless “angular arc length” is a misleading name. It is not any kind of an arc length. It would be really nice to give it some geometrical interpretation, but until now there is no such. Perhaps one day someone will invent such. That would be nice.

  3. Excellent post!!! The Wikipedia article makes no sense, and I found your post by searching for the nonsensical term “angular arc length.” You clarified all my questions.

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