Jacobi Elliptic Functions and the
Complete Solution to the Bead on the Hoop Problem

We start by redefining the basic trigonometric functions sine and cosine in terms of the functional inverse of specific integrals. Usually, these functions are introduced using circular geometry. That is, for a given point $(x,y)$ on a circle of radius $r$, $\sin\theta=y/r$ and $\cos\theta=x/r$. An alternative way to define these functions is to realize that each inverse trigonometric function is a solution of a definite integral. For example, simple trigonometric substitutions lead to

with the conditions that $-1\leq y\leq 1$ and $\sqrt{1-t^{2}}\geq 0$. We can reinterpret these relations by stating that they define the inverse trigonometric functions. Following Abel we obtain trigonometric functions by inverting these integrals.Prasolov97 For example, defining the argument $\theta$ as

we obtain the trigonometric sine function $\sin\theta=y$. With the Pythagorean relation, we can define the cosine function and all others follow. This alternative definition implies that the integrals $\theta(y)$ must in particular display the parity and periodicity of the corresponding trigonometric functions $y(\theta)$.

Consider now the case of elliptic geometry. Gauss, Legendre, Abel, Jacobi and Weierstrass studied in depth properties of elliptic functions and integrals. For the purpose of this paper we note only the result, proved by Legendre,Baker1906 that any integrated expression containing a third or fourth degree polynomial in the denominator of a fraction can be reduced to a linear combination of the following elliptic integrals of the first, second, and third kind,

where $y=\sin\phi$, $t=\sin\phi^{\prime}$, $k\in(-1,1)$ is the modulus and $n\in\mathbb{R}$ the characteristic. We assume all square roots to be positive. In Eqs. (II.1)–(II.1), the first expression (in $t$) is known as the Jacobi form whereas the second (in $\phi^{\prime}$) is Legendre’s form. When $\phi=\pi/2$, written $F\left(\tfrac{\pi}{2},k\right)\equiv K\left(k\right)$, the integrals are said to be complete and otherwise incomplete. We will use the first two elliptic integrals exclusively in what follows.

Because trigonometric functions are sometimes more useful than their inverses, it may be worthwhile to invert the elliptic integrals as well.Bowman53 ; Prasolov97 In this process, Abel and Jacobi followed the work of Legendre and introduced what are now known as the Jacobi elliptic functions. These functions result from the inversion of the elliptic integral of the first kind. Introducing the Jacobi notation $u(\phi)\equiv F(\phi,k)$ for this integral, we can formally invert the Legendre form of the integral, Eq. (II.1), to obtain $\phi(u)=\mathrm{am}\,u$, the amplitude of $u$. In the Jacobi form, $u(y)=F(y,k)$, the inversion of Eq. (II.1) leads one to define the sine-amplitude Jacobi elliptic function written in Glaisher’s notation ($y\in[-1,1]$, $\phi\in[-\pi/2,\pi/2]$)

Note that the $k$-dependence is often implicit in the literature, resulting in the notation $\mathrm{sn}\,u$ instead of $\mathrm{sn}\left(u,k\right)$. The last two equalities in Eq. (6) relate the Legendre and Jacobi forms of the elliptic integrals and justify the name given to this Jacobi elliptic function. Because Eq. (II.1) is odd, $u(y)=F(y,k)=\mathrm{sn}^{-1}(y,k)$ is odd in $y$ as well. That is, the sine-amplitude is an odd function of $u$.

It is natural to introduce two further Jacobi elliptic functions. Taking the cosine of the amplitude of $u$, we define the cosine-amplitude

An alternative way to define this function is to write $\mathrm{cn}\,u\equiv\sqrt{1-\mathrm{sn}^{2}u}$ since, from Eq. (6), $y\in[-1,1]$.

Finally, in the Legendre form all elliptic integrals in Eqs. (II.1)–(II.1) contain $\Delta(\phi)\equiv\sqrt{1-k^{2}\,\sin\phi}$. This leads one to define the delta-amplitude JEF that can also be written as a derivative of the amplitudeNeville71

Equation (II.1) can also be used to obtain the last equality. Figure 1 displays the characteristic behavior of the sine-, cosine- and delta-amplitude functions.

From the definitions above immediately follow the identities

For limiting values of $k$, the Jacobi elliptic functions reduce to trigonometric and hyperbolic functions

Jacobi elliptic functions thus include trigonometric and hyperbolic functions as special cases. However, JEFs are more than a simple generalization of elementary functions as can be seen when studying the properties of these functions in the complex plane (see Appendix A).

Finally, we mention that from the definitions (6)–(8), one can define quotients of JEFs. One defines $\mathrm{sc}\,u=\mathrm{sn}\,u/\mathrm{cn}\,u$ which carries the name tangent-amplitude $\mathrm{sc}\,u\equiv\mathrm{tn}\,u$. Other quotients are, for example, $\mathrm{nd}\,u=1/\mathrm{dn}\,u$, $\mathrm{sd}\,u=\mathrm{sn}\,u/\mathrm{dn}\,u$, etc. Further properties of JEFs that are used in the treatment of the bead on the hoop are summarized in Appendix A.

In the case of trigonometric functions, such as $y=\sin\theta$, the argument $\theta$ has an unequivocal geometric interpretation in circular geometry; it is the angle between the $x$-axis and the radius at a specific point $(x,y)$ on the circle. The question arises whether in a JEF the argument $u$ and the modulus $k$ have a geometric meaning as well. We first caution that the argument of a special function (Bessel, Hypergeometric, etc.) does not generally have such unequivocal physical or geometric meaning. For JEFs, the geometric definition of the argument $k$ is unique but not for the argument $u$, an important point not discussed explicitly in the literature.

The modulus $k$ is given in elliptic geometry by $k=\epsilon$, the eccentricity of the ellipse $\epsilon^{2}=1-b^{2}/a^{2}$, where $a>b$ are the semi-axes (see Fig. 2)

The equivalence between $k$ and $\epsilon$ is found by realizing that the arc length $s_{BP}$ on the ellipse is expressed in terms of the elliptic integral of the second kind. Using the parametrization $x=a\,\sin\phi$, $y=b\,\cos\phi$, the infinitesimal arc length along the ellipse is

Using Eq. (II.1), the arc length $s_{BP}$ of the ellipse from $B$ to $P$ can be written

This relation between the arc length of the ellipse and the function $E(\phi,k)$ is the reason why Fagnano (cited in Ref. Prasolov97, ) called $F$, $E$, and $\Pi$ elliptic integrals and provides a direct interpretation of the modulus of the JEFs.

The geometric interpretation of JEFs’ argument $u$ is less direct and not unique. Fagnano discovered that for $k^{-1}=\sqrt{2}$, the elliptic integral of the first kind describes the arc length on the lemniscate defined by the relation $r^{2}=\cos 2\theta$ (using the same definitions of $r$ and $\theta$ as in Fig. 2).Prasolov97 ; Armitage06 Serret, building on the work of Fagnano and Legendre, found a set of curves on which $F(\phi,k)$ at arbitrary value of $k\in(-1,1)$ is an arc length (see, e.g. Ref. Prasolov97, ). There is also a counterpart in three dimensions: Seiffert’s spirals defined on a sphere. The arc length of the spiral is given by the first elliptic integral.Erdos00 In these examples the argument $u$ of the JEF is the length of the Serret curve or the Seiffert spiral.

Additional geometric interpretations of the argument $u$ can be found from trigonometry as it applies to an ellipse using geodesy and elliptic geometry. This is discussed in detail in Refs. Bowman53, and bakerMS12, . All of these interpretations relate $u$ to a length of some geometric form related to an ellipse. It is important, however, to realize that the angles used to define $u$ are different for each physical situation. This leads to the important insight that the argument $u$ has no unique geometric interpretation beyond being an arc length and only has a physical interpretation in some special cases.Bowman53 ; Armitage06 ; bakerMS12

We begin by multiplying Eq. (20) by $\dot{\theta}$ and integrating to obtain

where the constant ${\cal C}$ is given by

and $\theta_{s}$ is given in Eq. (19). Note that Eq. (21) is not valid for $\omega=0$; this situation must be considered separately (see Sec. III.2.3). It is worth mentioning that $\theta_{s}$ is nonzero only for $\omega>\omega_{c}$ and takes values in the interval $\theta_{s}\in[0,\pi/2]$. For $\omega<\omega_{c}$, only $\theta_{0}=0$ is a stable stationary solution. It is then natural to extend the definition of $\theta_{s}$ by imposing $\theta_{s}=0$ for the case where $\omega<\omega_{c}$.

Because ${\cal C}$ is a constant, the left hand side of Eq. (21) is a constant of motion of the system. The expression for ${\cal C}$ is of particular interest because it allows us to reduce arbitrary initial conditions to ones with zero initial speed. Consider arbitrary initial conditions $(\tilde{\theta}_{0},\dot{\tilde{\theta}}_{0})$ for which $\dot{\tilde{\theta}}_{0}\neq 0$ and $E_{\mathrm{tot}}<V_{\mathrm{eff}}^{\mathrm{max}}$. From Eq. (21) we can find a time $t_{0}$ and new initial conditions $(\theta_{0},\dot{\theta}_{0})$ with zero initial angular speed $\dot{\theta}_{0}=0$ such that $\theta(t_{0})=\tilde{\theta}_{0}$ and $\dot{\theta}(t_{0})=\dot{\tilde{\theta}}_{0}$. The new initial condition $(\theta_{0},0)$ written in terms of the actual one $(\tilde{\theta}_{0},\dot{\tilde{\theta}}_{0})$ is

Thus, arbitrary initial conditions with $\dot{\tilde{\theta}}_{0}\neq 0$ can be written in terms of initial conditions with $\dot{\theta}_{0}=0$. In the following, we limit our discussion to the motion of the bead on the hoop with zero initial speed unless $E_{\mathrm{tot}}>V_{\mathrm{eff}}^{\mathrm{max}}$. This case will be discussed later.

We now use the Weierstrass substitutionSpivak06

in Eq. (21) and obtain

through separation of variables. This type of equation is usually studied numerically or in limiting cases only.Whittaker37 ; Lima10 ; Bowman53 Here we extend the calculation and invert the integral to obtain $\theta(t)$. The structure of the integral implies that the inverse is expressed in terms of JEFs defined in Sec. II.

For convenience, we define $\tau_{0}=\tan\left(\theta_{0}/2\right)$ and the two constants

After factoring out $p_{-}$ from the denominator of the integrand in Eq. (25), we write the remaining polynomial under the square root in the form

where the roots are given by

with

The root $a$ is always non-negative while $b$ can take any real value. As a result, the relative values of the two roots $a$ and $b$ divides the range $\theta_{0}\in[0,\pi]$ into one (for $\omega\leq\omega_{c}$) or two (for $\omega>\omega_{c}$) intervals. When $\omega>\omega_{c}$, the two intervals are divided by the point $b=0$, which occurs when the initial condition $\theta_{0}$ takes the value

For $\theta_{0}<\theta_{e}$ we have $b>0$ while for $\theta_{0}>\theta_{e}$ we have $b<0$. The angle $\theta_{e}$ appears in a natural way in our mathematical analysis and is new in the context of the bead and the hoop problem. This angle is represented with a star in Figs. 3 and 4.

We call $\theta_{e}$ the “ejection angle” because for any $\theta_{0}<\theta_{e}$, the bead will remain on the initial half of the hoop forever and never reach the other side. However, when $\theta_{0}>\theta_{e}$ the bead has sufficient energy to cross the relative maximum at $\theta=0$ and eject itself from the original half of the hoop where it was released. We note that $\theta_{e}(\omega_{c})=\theta_{s}(\omega_{c})=0$ which compels us to extend their definition by imposing $\theta_{e}=\theta_{s}=0$ for $\omega\leq\omega_{c}$. The two angles $\theta_{s}$ and $\theta_{e}$ also satisfy the inequality $\theta_{e}\geq\theta_{s}$ and $\theta_{e}\in[0,\pi]$ where $\theta_{e}(\omega\to\infty)\to\pi$. Curiously, we observe that $V_{g}\left(\theta_{e}\right)=2V_{g}\left(\theta_{s}\right)$ but are unaware if this is the result of some physical constraint. Further considerations where the bead is placed directly on this point are addressed in Sec. III.2.1.

The integral Eq. (25), along with Eq. (27), can be related to the first elliptic integral [Eq. (II.1)]. Inversion of the integral gives $\tau(t)$ and thus $\theta(t)$ using Eq. (24). Which particular inverse JEF equals Eq. (25) depends on $\gamma_{1}$ and $\gamma_{2}$ (or $a$ and $b$).

For $\omega\leq\omega_{c}$, we have $\theta_{s}=\theta_{e}=0$ and $a=b\geq 0$ for $\theta_{0}\in[0,\pi]$. From the tables in Ref. Byrd71, with $\gamma_{1}^{2}=a$ and $\gamma_{2}^{2}=b$ in Eq. (27), we identify the cosine-amplitude. Transforming back to the variable $\theta=2\arctan\tau$, we obtain

On the other hand, when $\omega>\omega_{c}$ the range $\theta_{0}\in[0,\pi]$ is divided into three intervals $[0,\theta_{s}]$, $[\theta_{s},\theta_{e}]$, and $[\theta_{e},\pi]$, each of which leads to a solution involving a different JEF. For $\theta_{0}\in\left[0,\theta_{s}\right]$ the roots obey the inequality $b>a\geq 0$; for $\theta\in\left[\theta_{s},\theta_{e}\right]$ we have $a>b\geq 0$; and for $\theta\in[\theta_{e},\pi]$ we have $a\geq 0\geq b$. Using again the table of integrals in Ref. Byrd71, we set $\gamma_{1}^{2}=a$ and $\gamma_{2}^{2}=b$ for regions $\theta_{0}\in\left(0,\theta_{s}\right)$ and $\theta_{0}\in\left(\theta_{e},\pi\right)$, whereas $\gamma_{1}^{2}=b$ and $\gamma_{2}^{2}=a$ for $\theta_{0}\in\left(\theta_{s},\theta_{e}\right)$. This allows identifying the Jacobi elliptic functions nd, dn, and cn, respectively. The third interval $\theta_{e}\leq\theta\leq\pi$, turns out to lead to an expression that is identical to Eq. (31) above.

One can reduce the three cases for $\omega>\omega_{c}$ to one single expression for the entire interval $[0,\pi]$ by using properties of the JEFs derived in Sec. II.1 and Appendix A. The simplest form of the solution is found in terms of the delta-amplitude function, which naturally appears in the second interval. The cosine-amplitude function appearing in Eq. (31) can be transformed into a delta-amplitude with Eq. (45). Similarly, using the relation discussed after Eq. (46), $k^{\prime}\,\mathrm{nd}(u,k)=\mathrm{dn}(u+K,k)$, it is possible to reduce the ${\rm nd}$ solution in the first interval to a delta-amplitude function. Careful consideration of the argument $(u,k)$ in the transformation leads to the final result

This is the exact solution of the equation of motion (20), which fully describes the motion of a bead on a spinning hoop for arbitrary initial conditions $(\theta_{0},\dot{\theta}_{0})$, excepting the cases where $E_{\mathrm{tot}}\geq V_{g}(\pi)$ or $\omega=0$. Equation (32) simplifies to $\theta(t)=\theta_{s}$ for all times if $\theta_{0}=\theta_{s}$. This solution improves or complements previous studies of different variants of the problem.Taylor05 ; Ochoa06 ; Lima10 ; Rousseaux09 ; Bowman53 ; Whittaker37

We note that the argument $u(t)$ of the JEF is a linear function of time and is proportional to $\phi=\omega t$ given by Eq. (25). As pointed out in Sec. II, it is only in special cases that the argument has a useful physical interpretation, and the bead on the hoop is not one of them. In fact, the problem is slightly more difficult than what was presented in the previous section because the JEF was only identified after performing a Weierstrass substitution on the integral of Eq. (25). The Weierstrass substitution can be shown to be a particular mapping of the position of the bead onto the $xy$-plane. The map is obtained from the intersection between the $xy$-plane and the line passing through the top of the hoop and through the bead. The argument $u(t)$ is related to the arc length $(R\sqrt{|p_{-}|\,a}/2)\,\phi$ on this stereographic projection.bakerMS12

We discuss the special cases not considered in the previous section: the separatrices ($E=0,\,2mgR$), the free spinning of the bead around the hoop ($E>2mgR$), and the general solution for the pendulum ($\omega=0$). For all these cases the roots (29) take values $a,b\to\infty$ or 0 and the determination of $\theta(t)$ requires special care.