Calculation of Elliptic Functions at Extreme Values of Eccentricity

Question

The scenario of transverse oscillations of a mass in centre of an elastic string yields an oscillator with a restoring force that is purely cubic ... or a potential well that is quartic. The solution of the transverse motion as a function of time is the Jocobi elliptic function $\operatorname{sn}$ with eccentricity $k=i \therefore k^2=-1$. I wondered what method of computation can still be used at this value of $k$ - and that is infact the question.

In the method of Neville theta functions, the nome

$$q=\exp(-\pi K^\prime/K)$$

is required; but now, since the complementary eccentricity-squared $k^{\prime 2}=1-k^2=2$ we will have a complex

$$K^\prime=\int_0^1\frac{dx}{\sqrt{(1-2x^2)(1-x^2)}} .$$

I should expect that $sn(\omega t)$ would be a quite reasonable real-valued function of $\omega t$, as the solution (with all the scalings taking out) is simply the inverse of

$$\int_0^x\frac{d\upsilon}{\sqrt{1-x^4}} ,$$

which would basically be a $\sin$ curve a bit 'fattened-out'. But do the imaginary parts cancel-out in the expansion the expansions comprised in a Neville's theta function rendering of it? And also, if it be rendered as a Taylor series instead, will this series converge? It is generally said that a condition for convergence is $|k|<1$ ... and if it does converge it is likely that it will be very slowly ... and calculating an arbitrary number of coefficients of the series for $sn$ is not a trivial matter (have you ever seen the recipe for them!!?).

In practice, people are not particularly interested in calculating the trajectory of a quartic oscillator - they generally want just to 'short-circuit' straight to the energy-levels, & that kind of thing ... and furthermore, they might adduce that it has already been done once & forall anyway. But! ... I always want to be able to do things like that myself - I supppose it is a bit of a vice that I have!

So I wonder at what sort of values of $k$ there is a transition from the use of these elementary methods that you might well find in old treatises to the slick algorithms that I've no doubt the great computer algebra packages have on board. And there are subquestions implied in the main one, such as "what is the rate of growth, wrt index, of the sum of the coefficients of the polynomials in k^2 that constitute the numerators of the coefficients of the Taylor series for $\operatorname{sn}$?". It doesn't appear, from my perusals of the literature anyway, that this question, for instance, is much-addressed.

The inspiration for this post further back was actually materials with negative thermal expansion coefficients: some of them do not readily evince the precise mechanism whereby they have this property; but in a notable one of them - scandium trifloride - it is that the atoms are arranged in such a way that a flouride ion, undergoing transverse quartic oscillations between two scandium ions tends to draw the scandium ions closer together ... which I found really neat!

I am also led to wonder what physicalities $k^2<-1$ might have: I noticed that -1 is a sort of watershed beyond which $\operatorname{sn}$ begins to acquire a dent (not kink, as I said at first, which maybe implies a discontinuity of gradient) - which is to say, being more precise, its gradient actually increases just after a zero, before levelling-off to turn back. The uttermost extreme of this would be $k^2\rightarrow-\infty$ whereat (provided our scale of observation decrease in proportion with $|k^2|$) it morphs into a $\operatorname{sech}$ function ... if it still be taken as the inverse of the elliptic integral of the first kind.

Answer 1

I've got it now: that the solution is (using traditional notation for elliptic integrals/functions - ie that the parameter is $k$, not $k^2$)

$$\operatorname{sn}(z,ik)$$$$=\frac{k\operatorname{sn}\left(\sqrt{\frac{1}{k^2}+1}z, \frac{k}{\sqrt{1+k^2}}\right)}{\sqrt{1+k^2\operatorname{cn^2}(\sqrt{\frac{1}{k^2}+1}}z, \frac{k}{\sqrt{1+k^2}})} ,$$

which for $k=1$ gives

$$\operatorname{sn}(z,i)$$$$=\frac{\operatorname{sn}\left(\sqrt{2}z, \frac{1}{\sqrt{2}}\right)}{\sqrt{1+\operatorname{cn}^2(\sqrt{2}}z, \frac{1}{\sqrt{2}})} ,$$

$$\operatorname{sn}(z,i)$$$$=\frac{\operatorname{sn}\left(\sqrt{2}z, \frac{1}{\sqrt{2}}\right)}{\sqrt{2-\operatorname{sn^2}(\sqrt{2}}z, \frac{1}{\sqrt{2}})} .$$

I have also seen

$$\operatorname{sn}(z,i)$$$$=\operatorname{cn}\left(K\left(\frac{1}{\sqrt{2}}\right)-\sqrt{2}z, \frac{1}{\sqrt{2}}\right) ,$$

but I do not know exactly from what transformations & identities this is obtained. I have also read that this function has been given a special place amongst the canon of functions (similarly to how $\operatorname{gd}x\equiv\operatorname{ln}\operatorname{tan}(\frac{\pi}{4}+\frac{x}{2})$ sometimes is), being called sinus lemniscatus or $\operatorname{sl}$; and is, as a function of time, the distance from the origin of a point moving with unit speed along a lemniscate.

Also, for the purpse of dedimensionalising the problem: if we set $q$ the force-constant (force per (unit length)$^3$), & $m$ mass, & $v_0$ the speed at zero displacement, then

$$\Upsilon=\sqrt{\sqrt{\frac{2m}{q}}}$$

has dimensions $\sqrt{LT}$; & the characteristic length is $$\Upsilon\sqrt{v_0}$$ & the characteristic time is $$\frac{\Upsilon}{\sqrt{v_0}} .$$

I've also found out that the Taylor series for $\operatorname{sn}$ is divergent at unit argumentsize, which means that the sum of the coefficients must increase at least factorially - that being a watershed - a series being convegent at unit-argument when the factorial of the numerator is retarded by more than 1 with respect to that of the denominator, and convergent if it is retarded by exactly 1 or aligned or atall advanced; and the series convergent at unit argument if the numerator is atall subfactorial.

It's also fascinating that whereas the solution of the quartic oscillator problem can be expressed as the distance-from-origin of a point moving with unit speed along a lemniscate $r=\cos\frac{\theta}{2}$, the solution for an ordinary quadratic oscillator can likewise be expressed as the distance-from-origin of a point moving with unit speed along a fattened 'lemniscate' $r=\cos\theta$ ... or two circles just fitting within the unit circle.

I can't offhand think of how a hektic oscillator might simply physically be realised.