The last post, “Dzhanibekov effect – Part 5 – the Navy link“, ended with:

But we will need Jacobi elliptic functions, and so in a couple of forthcoming posts I will have to post a mini-tutorial on the subject. …

So, today I will start this “mini-tutorial”. Though I must say that it is not really necessary for understanding the physics of the flipping nut or a tennis racket. Yes, we will need some elliptic functions, but you do not have to know how they are defined or how they are computed. It is like with paying in a shop: you need the money, but you do not need to know when and how the money has been produced. Well, you need to know that it is real, not Counterfeit, but usually you do not pay any attention at all. The same is with elliptic functions that we need. Computer software nowadays is smart enough to know how to calculate elliptic functions, and Wikipedia is always there to give us some idea about what they are and to suggest some real good sources. Nevertheless why do not impress friends (and enemies as well) by telling them: I know what elliptic sinus is! It may be a nice conversation subject at a party!

And so we turn to Jacobielliptic functions. Jacobi was an interesting person. According to Wikipedia:

Carl Gustav Jacob Jacobi (/dʒəˈkoʊbi/;[1] German: [jaˈkoːbi]; 10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.
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Jacobi was the first Jewish mathematician to be appointed professor at a German university.
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He followed immediately with his Habilitation and at the same time converted to Christianity.
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In 1827 he became a professor and in 1829, a tenured professor of mathematics at Königsberg University, and held the chair until 1842.
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Jacobi suffered a breakdown from overwork in 1843.
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Jacobi died in 1851 from a smallpox infection.

The book “Remarkable Mathematicians From Euler to von Neumann” by Ioan James has a whole chapter about Jacobi. There we learn, in particular, that the weather in Königsberg was very severe, and it has affected Jacobi’s health. Königsberg, Immanuel Kant’s town, then Prussia, now in Russia, perhaps is not conductive to mathematics, but, according to Russian Forbes, is one of the best Russian towns for conducting business….

Anyway, Jacobi introduced what we call Jacobi elliptic functions. We will need some of them, in particular the functions sn, cn and dn. The function sn is a generalization of the ordinary sinus. In fact it is an interbreed between trigonometric and hyperbolic sinus. Similarly cn is a deformation of the ordinary cosinus, while dn is …. a deformation of 1. I am going to introduce these functions following the ideas given by William Schwalm in his handout notes “Elliptic Functions sn, cn, dn, as Trigonometry“. There is als a book by William A. Schwalm, “Lectures on Selected Topics in Mathematical Physics: Elliptic Functions and Elliptic Integrals“, with the first chapter entitled “Elliptic functions as trigonometry”. I am not yet completely happy with this particular way of introducing these functions, so this morning I wrote to prof. Schwalm kindly asking for additional explanations. If I get an answer, I will tell you what it is about.
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Well, I got an answer almost immediately. It introduced me to some kind of a “war” between prof. Schwalm and one “Wikipedia librarian”. Wikipedia also has “Definition as trigonometry” section in Jacobi Elliptic Functions entry. I have to yet figure out what it is about. So, please, be patient. I will continue in the next post.