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\author{Raph Levien}
\title{The elastica: a mathematical history}

\maketitle

\abstract

This report traces the history of the elastica from its first precise
formulation by James Bernoulli in 1691 through the present. The
complete solution is most commonly attributed to Euler in 1744 because
of his compelling mathematical treatment and illustrations, but in
fact James Bernoulli had arrived at the correct equation a
half-century earlier. The elastica can be understood from a number of
different aspects, including as a mechanical equilibrium, a problem of
the calculus of variations, and the solution to elliptic integrals. In
addition, it has a number of analogies with physical systems,
including a sheet holding a volume of water, the surface of a
capillary, and the motion of a simple pendulum. It is also the
mathematical model of the mechanical spline, used for shipbuilding and
similar applications, and directly insipired the modern theory of
mathematical splines. More recently, the major focus has been on
efficient numerical techniques for computing the elastica and fitting
it to spline problems. All in all, it is a beautiful family of curves
based on beautiful mathematics and a rich and fascinating history.

This report is adapted from a Ph. D. thesis done under the direction
of Prof. C. H. S\'equin.

\section{Introduction}

This report tells the story of a remarkable family of curves, known as
the elastica, Latin for a thin strip of elastic material. The elastica
caught the attention of many of the brightest minds in the history of
mathematics, including Galileo, the Bernoullis, Euler, and others. It
was present at the birth of many important fields, most notably the
theory of elasticity, the calculus of variations, and the theory of
elliptic integrals. The path traced by this curve illuminates a wide
range of mathematical style, from the mechanics-based intuition of the
early work, through a period of technical virtuosity in mathematical
technique, to the present day where computational techniques dominate.

There are many approaches to the elastica. The earliest (and most
mathematically tractable), is as an equilibrium of moments, drawing on
a fundamental principle of statics. Another approach, ultimately
yielding the same equation for the curve, is as a minimum of bending
energy in the elastic curve. A force-based approach finds that normal,
compression, and shear forces are also in equilibrium; this approach
is useful when considering specific constraints on the endpoints,
which are often intuitively expressed in terms of these forces.

Later, the fundamental differential equation for the elastica was
found to be equivalent to that for the motion of the simple
pendulum. This formulation is most useful for appreciating the curve's
periodicity, and also helps understand special values in the parameter
space.

In more recent times, the focus has been on efficient numerical
computation of the elastica (especially for the application of fitting
smooth spline curves through a sequence of points), and also
determining the range of endpoint conditions for which a stable
solution exists. Many practical applications continue to use
brute-force numerical techniques, but in many cases the insights of
mathematicians (many working hundreds of years earlier) still have
power to inspire a more refined computational approach.

%These days, the most familiar and direct understanding of the elastica
%would probably be as the limit to a finite element problem.

\section{Jordanus de Nemore---13th century}

In the recorded literature, the problem of the elastica was first
posed in \emph{De Ratione Ponderis} by Jordanus de Nemore (Jordan of
the Forest), a thirteenth century mathematician. Proposition 13 of
book 4 states that ``when the middle is held fast, the end parts are
more easily curved.'' He then poses an incorrect solution: ``And so it
comes about that since the ends yield most easily, while the other
parts follow more easily to the extent that they are nearer the ends,
the whole body becomes curved in a circle.'' \cite{DAntonio07}. In
fact, the circle is one possible solution to the elastica, but not to
not for the specific problem posed. Even so, this is a clear statement
of the problem, and the solution (though not correct) is given in the
form of a specific mathematical curve. It would be several centuries
until the mathematical concepts needed to answer the question came
into existence.

\section{Galilieo sets the stage---1638}

Three basic concepts are required for the formulation of the elastica
as an equilibrium of moments: moment (a fundamental principle of
statics), the curvature of a curve, and the relationship between these
two concepts. In the case of an idealized elastic strip, these
quantities are linearly related.

\begin{figure*}[htb]
\begin{center}
\includegraphics[width=3in]{figs/galileo.png}
\caption{\label{galileo}Galileo's 1638 problem.}
\end{center}
\end{figure*}

Galileo, in 1638, posed a fundamental problem, founding the
mathematical study of elasticity. Given a prismatic beam set into a
wall at one end, and loaded by a weight at the other, how much weight
is required to break the beam? The delightful Figure \ref{galileo}
illustrates the setup. Galileo considers the beam to be a compound
lever with a fulcrum at the bottom of the beam meeting the wall at
B. The weight E acts on one arm BC, and the thickness of the beam at
the wall, AB, is the other arm, ``in which resides the resistance.''
His first proposition is, ``The moment of force at C to the moment of
the resistance... has the same propotion as the length CB to the half
of BA, and therefore the absolute resistance to breaking... is to the
resistance in the same proportion as the length of BC to the half of
AB...''

From these basic principles, Galileo derives a number of results,
primarily a scaling relationship. He does not consider displacements
of the beam; for these types of structural beams, the displacement is
neglible. Even so, this represents the first mathematical treatment of
a problem in elasticity, and firmly establishes the concept of moment
to determine the force on an elastic material. Many researchers
elaborated on Galileo's results in coming decades, as described in
detail in Todhunder's history \cite{Todhunter}.

One such researcher is Ignace-Gaston Pardies, who in 1673 posed one
form of the elastica problem and also attempts a solution: for a beam
held fixed at one end and loaded by a weight at the other, ``it is
easy to prove'' that it is a parabola. However, this solution isn't
even approximately correct, and would later be dismissed by James
Bernoulli as one of several ``pure fallacies.''
Truesdell gives Pardies credit for
introducing the elasticity of a beam into calculation of its
resistance, and traces out his influence on subsequent researchers,
particularly Leibniz and James Bernoulli \cite[p. 50--53]{Truesdell60}.

\section{Hooke's law of the spring---1678}

Hooke published a treatise on elasticity in 1678, containing his
famous law. In a short Latin phrase posed in a cryptic
anagram three years earlier to establish priority, it reads,
``\emph{ut tensio sic vis}; that is, The Power of any Spring is in the
same proportion with the Tension thereof... Now as the Theory is very
short, so the way of trying it is very easie.'' (spelling and
capitalization as in original). In modern formulation,
it is understood as $F \propto \Delta l$; the applied force is
proportional to the change in length.

\begin{figure*}[htb]
\begin{center}
\includegraphics[width=2in]{figs/hooke.png}
\caption{\label{fig-hooke}Hooke's figure on compound elasticity.}
\end{center}
\end{figure*}


Hooke touches on the problem of elastic strips, providing an evocative
illustration (Figure \ref{fig-hooke}), but according to Truesdell,
``This `compound way of springing' is the main problem of elasticity
for the century following, but Hooke gives no idea how to relate the
curvature of one fibre to the bending moment, not to mention the
reaction of the two fibres on one another.'' \cite[p. 55]{Truesdell60}

Indeed, mathematical understanding of curvature was still in
development at the time, and mastery over it would have to wait for
the calculus. Newton published results on curvature in his ``Methods
of Series and Fluxions,'' written 1670 to 1671 \cite[p. 232]{Harman02}, but not published for
several more years. Leibniz similarly used his competing version of
the calculus to derive similar results. Even so, some results were
possible with pre-calculus methods, and in 1673, Christiaan Huygens
published the ``Horologium oscillatorium sive de motu pendulorum ad
horologia aptato demonstrationes geometrica,'' which used purely
geometric constructions, particularly the involutes and evolutes of
curves, to establish results involving curvature. The flavor of
geometric construction pervades much of the early work on elasticity,
particularly James Bernoulli's, as we shall see.

Newton also used his version of the calculus to more deeply understand
curvature, and provided the formulation for radius of curvature in
terms of Cartesian coordinates most familiar to us today
\cite{Harman02}:

\begin{equation}
\label{newton-curvature-eq}
\rho = (1 + y'^2)^{\frac{3}{2}}/y''
\end{equation}

An accessible introduction to the history of curvature is the aptly
named ``History of Curvature'' by Dan Margalit \cite{Margalit03}.

Given a solid mathematical understanding of curvature, and assuming a
linear relation between force and change of length, working out the
relationship between moment and curvature is indeed ``easie,'' but
even Bernoulli had to struggle a bit with both concepts. For one,
Bernoulli didn't simply accept the linear law of the spring, but felt
the need to test it for himself. And, since he had the misfortune to
use catgut, rather than a more ideally elastic material such as metal,
he found significant nonlinearities.
Truesdell \cite{Truesdell87} recounts a letter
from James Bernoulli to Leibniz on 15 December 1687, and the reply of
Leibniz almost three years later. According to Truesdell, this
exchange is the birth of the theory of the ``curva elastica'' and its
ramifications. Leibniz had proposed: ``From the hypothesis elsewhere
substantiated, that the extensions are proportional to the stretching
forces...'' which is today attributed as Hooke's law of the
spring. Bernoulli questioned this relationship, and, as we shall see,
his solution to the elastica generalized even to nonlinear displacement.

\section{James Bernoulli poses the elastica problem---1691}

James Bernoulli posed the precise problem of the elastica in 1691:

\begin{figure*}[htb]
\begin{center}
\includegraphics[width=2in]{figs/bernoulli1691.png}
\caption{\label{fig-elastica1691}Bernoulli poses the elastica problem.}
\end{center}
\end{figure*}

``Si lamina elastica gravitatis espers AB, uniformis ubique crassitiei \&
latitudinis, inferiore extremitate A alicubi firmetur, \& superiori B
pondus appendatur, quantum sufficit ad laminam eousque incurvandam, ut
linea directionis ponderis BC curvatae laminae in B sit
perpendicularis, erit curvatura laminae sequentis naturae:''

And then in cipher form:

``Portio axis applicatam inter et tangentem est ad ipsam tangentem sicut
quadratum applicatae ad constans quoddam spatium.''\footnote{The cipher
  reads ``Qrzumu bapt dxqopddbbp...'' and the key was published in the
  1694 \emph{Curvatura Laminae} with the detailed solution to the
  problem. Such techniques for establishing priority may seem alien to
  academics today, but are refreshingly straightforward by comparison to
  the workings of the modern patent system.}

%Si lamina elastica gravitatis espers AB, uniformis ubique crassitei \&
%if lamina elastic weight free from AB, uniform everywhere thickness and
%
%latitudinis, inferiore extremitate A alicubi firmetur, \& superiori B
%width, below perimeter A somewhere support, and above B
%
%pondus appendatur, quantum sufficit ad laminam eousque incurvandam, ut
%weight hang, how much suffice to lamina ? cause to bend down, to
%
%linea directionis ponderis BC curvatae laminae in B sit
%line directing weight BC curved lamina in B is
%
%perpendicularis, erit curvatura laminae sequentis naturae:
%perpendicular, is curve lamina follows by nature:

\begin{quote}
Assuming a lamina AB of uniform thickness and width and negligible
weight of its own, supported on its lower perimeter at A, and with a
weight hung from its top at B, the force from the weight along the
line BC sufficient to bend the lamina perpendicular, the curve of the
lamina follows this nature:

%Portio axis applicatam inter et tangentem est ad ipsam tangentem sicut
%portion axis apply between and tangent is about itself tangent as
%
%quadratum applicatae ad constans quoddam spatium.
%square apply about constant certain space.

The rectangle formed by the tangent between the axis and its own
tangent is a constant area.
\end{quote}

This poses one specific instance of the general elastica problem, now
generally known as the \emph{rectangular elastica}, because the force
applied to one end of the curve bends it to a right angle with the
other end held fixed.

The deciphered form of the anagram is hardly less cryptic than the
original, but digging through his 1694 explanation, it is possible to
extract the fundamental idea: at every point along the curve, the
product of the radius of curvature and the distance from the line BC
is a constant, i.e. the two quantities are inversely proportional.
And, indeed, that is the key to unlocking the elastica; the equation
for the shape of the curve follows readily, given sufficient
mathematical skill.

\section{James Bernoulli partially solves it---1692}

%Euler begins his chapter crediting the Bernoullis for opening the
%investigation of the elastica, and finding many solutions. Likely the
%earliest known record (see Truesdell's commentary in his introduction
%to a volume of James Bernoulli's correspondence \cite{Truesdell}) of
%the elastica equation is Meditation CXII, a 1692 manuscript from James
%Bernoulli, which begins thus:


\begin{figure*}[htb]
\begin{center}
\includegraphics[width=2in]{figs/bernoulli2.png}
\hspace{.5in}
\raisebox{0in}{\includegraphics[width=1.25in]{figs/bernoulli91.pdf}}
\caption{\label{bernoulli-clxx}Drawing from James Bernoulli's 1692 Med. CLXX, with modern reconstruction.}
\end{center}
\end{figure*}


By 1692, James Bernoulli had completely solved the rectangular case of the
elastica posed earlier. In his Meditatione CLXX dated that year, titled
\emph{``Quadratura Curvae, e cujus evolutione describitur inflexae
  laminae curvatura''} \cite{BernoulliJacCLXX}, or, ``Quadrature of a
curve, by the the evolution of which is traced out the curve of a bent
lamina'' he draws a figure of that curve (reproduced here as Figure
\ref{bernoulli-clxx}), and gives an equation for its
quadrature\footnote{Today, we would say simply ``integral'' rather
  than quadrature.}.

\begin{quote}
Radius circuli $AB = a$, $AED$ lamina elastica ab appenso pondere in
$A$ curvata, $GLM$ illa curva, ex cujus evolutione $AED$ describitur:
$AF = y$, $FE = x$, $AI = p$, $IL = z$. Aequatio differentialis
naturam curvae $AED$ exprimens,

\[
dy = \frac{xx\; dx}{\sqrt{a^4 - x^4}},
\]

ut suo tempore ostendam:
% \& quia $LI(z)$ invenitur =
%$\frac{1}{2}EF(\frac{1}{2}x)$, reperitur exinde aequatio naturam
%curvae $GLM$ exprimens
%
%\[
%\frac{dx\; dz}{dy} = dp = \frac{dz\sqrt{a^4 - 16z^4}}{4zz},
%\]
%
%adeoque differentiale spatii $GLIH$
%
%\[
%= z\; dp = \frac{z\; dz\; \sqrt{a^4 - 16z^4}}{4zz}.
%\]

\end{quote}

A rough translation:

\begin{quote}
Let the radius of the circle $AB = a$, $AED$ be an elastic lamina
curved by a suspended weight at $A$, and $GLM$ that curve, the
involute of which describes $AED$. $AF = y$, $FE = x$, $AI = p$, $IL =
z$. The differential equation for the curve $AED$ is expressed,

\begin{equation}
\label{bernoulli-elastica}
dy = \frac{x^2\; dx}{\sqrt{a^4 - x^4}},
\end{equation}

as I will show in due time.
% \& because $LI(z)$ discover =
%$\frac{1}{2}EF(\frac{1}{2}x)$, find then equation natural curve $GLM$ express
%
%\[
%\frac{dx\; dz}{dy} = dp = \frac{dz\sqrt{a^4 - 16z^4}}{4zz},
%\]
%
%indeed differential space $GLIH$
%
%\[
%= z\; dp = \frac{z\; dz\; \sqrt{a^4 - 16z^4}}{4zz}.
%\]

\end{quote}

Equation \ref{bernoulli-elastica} is a clear, simple statement of the
differential equation for the rectangular instance of the elastica
family, presented in readily computable form; $y$ can be obtained as
the integral of a straightforward function of $x$.

Based on the figure, $AED$ is the elastica, and $GLM$ is its evolute.
Thus, the phrase ``evolutione describitur'' appears to denote taking
the \emph{involute} of the curve $GLM$ to achieve the elastica $AED$;
$L$ is the point on the evolute corresponding to the point $E$ on the
elastica itself. $L$ has Cartesian coordinates $(p, z)$, likewise $E$ has
coordinates $(x, y)$. It is clear that Bernoulli was working with the
evolute due as the geometric construction for curvature,
which is of course central to the theory of the elastica. In
particular, by the definition of the evolute, the length $EL$ is the
radius of curvature at point $E$ on the curve $AED$.

%Of course, this equation captures only one instance of the general
%elastica problem, and Todhunter explains why \cite[p. 12]{Todhunter}:
%
%\begin{quote}
%The investigation considered as the solution of a mechanical problem
%is imperfect; we know that \emph{three} equations must be satisfied in
%order to ensure equilibrium among a set of forces in one plane, but
%here only \emph{one} equation is regarded, namely that of moments.
%\end{quote}
%
%Indeed, in the case of the rectangular elastica the compression force
%at A vanishes, so in this case an analysis based on moments alone
%suffices.

\section{James Bernoulli publishes the first solution---1694}

\begin{figure*}[htb]
\begin{center}
\includegraphics[width=5.5in]{figs/bernoulli1694.png}
\caption{\label{fig-elastica1694}Bernoulli's 1694 publication of the elastica.}
\end{center}
\end{figure*}

Bernoulli held on to this solution for a couple of years, and finally
published in his landmark 1694 \emph{Curvatura Laminae
  Elasticae}\footnote{\emph{Curvatura Laminae Elasticae. Ejus
    Identitas cum Curvatura Lintei a pondere inclusi fluidi
    expansi. Radii Circulorum Osculantium in terminis simplicissimis
    exhibiti, una cum novis quibusdam Theorematis huc pertinentibus,
    \&c.'',} or ``The curvature of an elastic band. Its identity with
  the curvature of a cloth filled out by the weight of the included
  fluid. The radii of osculating circles exhibited in the most simple
  terms; along with certain new theorems thereto pertaining,
  etc.''. Originally published in the June 1694 \emph{Acta Eruditorum}
  (pp. 262--276), it is collected in the 1744 edition of his
  \emph{Opera} \cite[p. 576--600]{BernoulliOpera1}, now readily
  accessible online.}. See Truesdell \cite[pp. 88--96]{Truesdell60} for
a detailed description of this work, which we will only outline here.

\begin{figure*}[htb]
\begin{center}
\includegraphics[width=2in]{figs/curvature.png}
\caption{\label{fig-curvature}Bernoulli's justification for his
  formula for curvature.}
\end{center}
\end{figure*}

Bernoulli begins the paper by giving a general equation for curvature
(illustrated in Figure \ref{fig-curvature}), which he introduces thus,
``in simplest and purely differential terms the relation of the evovent
of radius of the osculating circle of the curve... Meanwhile, since
the immense usefulness of this discovery in solving the velaria, the
problem of the curvature of springs we here consider, and other
recondite matters makes itself daily more and more manifest to me, the
matter stands that I cannot longer deny to the public the golden
theorem.'' Bernoulli's formulation is not entirely familiar to the
modern reader, as it mixes infinitesimals somewhat promiscuously. He
sets out this formula for the radius of curvature $z$:

\begin{equation}
\label{golden-theorem}
z = \frac{dx \, ds}{ddy} = \frac{dy\, ds}{ddx}
\end{equation}

From Bernoulli's tone, it is clear he thought this was an original
result, but Huygens and Leibniz were both aware of similar formulas;
Huygens had already published a statement and proof in terms of pure
Cartesian coordinates (which would be the form most familiar
today). However, in spite of this knowledge, both considered the
problem of the elastica impossibly difficult. Huygens, in a letter to
Leibniz dated 16 November 1691, wrote, ``I cannot wait to see what
Mr. Bernoulli the elder will produce regarding the curvature of the
spring. I have not dared to hope that one would come out with anything
clear or elegant here, and therefore I have never tried.'' \cite[p. 88,
  footnote 4]{Truesdell60}

Bernoulli's treatment of the elastica is fairly difficult going (as
evidenced by the skeptical reaction and mistaken conclusions from
Leibniz and others), but again, it is possible to tease out the
central ideas. First, the idea that the moment at any point along the
curve is proportional to the distance from the line of
force. Bernoulli writes in a 1695 paper
\footnote{ \emph{Explicationes,
    Annotationes et Additiones ad eas quae in actis superiorum annorum
    de curva elastica, isochrona paracentrua, et velaria, hinc inde
    memorata, et partim controversa leguntur; ubi de linea mediarum
    directionum, aliisque novis}. Originally published in
  Acta. Eruditorum, Dec. 1695, pp. 537-553, and reprinted in the 1744
  Complete Works \cite[pp. 639--663]{BernoulliOpera1}.} explaining the 1694
publication (and referring to Figure \ref{fig-elastica1694} for the
legend): ``I consider a lever with fulcrum $Q$, in which the thickness
$Qy$ of the band forms the shorter arm, the part of the curve $AQ$ the
longer. Since $Qy$ and the attached weight $Z$ remain the same, it is
clear that the force stretching the filament $y$ is proportional to
the segment $QP$.''  Next, Bernoulli carefully separates out the force
and the elongation, and, in fact, allows for a completely arbitrary
functional relationship, not necessarily linear (this function is
represented by the curve $AFC$ in Figure \ref{fig-elastica1694},
labelled ``Linea Tensionum''). ``And since the elongation is
reciprocally proportional to $Qn$, which is plainly the radius of
curvature, it follows that $Qn$ is also reciprocally proportional to
$x$.'' A central argument here is that the moment (and hence
curvature, assuming a linear relationship) is proportional solely to
the amount of force and the distance from the line of that force; the
shape of the curve doesn't matter.

\begin{figure*}[htb]
\begin{center}
\includegraphics[width=2in]{figs/moment_bernoul.pdf}
\caption{\label{fig-moment-bernoul}A simplified diagram of moments.}
\end{center}
\end{figure*}

The moments can be seen in the simplified diagram in Figure
\ref{fig-moment-bernoul}, which shows the curve of the elastica itself
(corresponding to $AQR$ in Figure \ref{fig-elastica1694}), the force
$F$ on the elastica (corresponding to $AZ$ in Figure
\ref{fig-elastica1694}), and the distance $x$ from the line of force
(corresponding to $PQ$ in Figure \ref{fig-elastica1694}). According to
the simple lever principle of statics, the moment is equal to the
force $F$ applied on the elastica times the distance $x$ from the
line of force.

Here we present a slightly simplified version of Bernoulli's argument
(see Truesdell \cite[p. 92]{Truesdell60} for a more complete
version). Write the curvature as a function of $x$, understanding that
in the idealized case it is linear, i.e. $f(x) = cx$. Then, using
Equation \ref{golden-theorem}, the ``golden theorem'' for curvature:

\begin{equation}
\frac{d^2y}{dx\, ds} = f(x)
\end{equation}

Integrating with respect to $dx$ and assuming $dy/ds = 0$ at $x = s = 0$,

\begin{equation}
\label{bernoulli-integral}
\frac{dy}{ds} = \int_0^x f(\xi) d\xi = S(x)
\end{equation}

Substituting the standard identity (which follows algebraically from
$ds^2 = dx^2 + dy^2$),

\begin{equation}
dy/dx = \frac{dy/ds}{\sqrt{1 - (dy/ds)^2}}
\end{equation}

we get:

\begin{equation}
\label{bernoulli-integrated}
\frac{dy}{dx} = \frac{S(x)}{\sqrt{1-S(x)^2}}
\end{equation}

And in the ideal case where $f(x) = 2cx$, we have $S(x) = cx^2$, and thus:

\begin{equation}
\frac{dy}{dx} = \frac{cx^2}{\sqrt{1-c^2x^4}}
\end{equation}

Which is the same as the Equation \ref{bernoulli-elastica}, with
straightforward change of constants. As mentioned before, the use of
both $ds$ and $dx$ as the infinitesimal is strange by modern
standards, even though in this case it leads to a solution quite
directly. A parallel derivation using similar principles but using
only Cartesian coordinates can be found in Whewell's 1833 Analytical
Statics \cite[p. 128]{Whewell1833}; there, Bernoulli's use of $ds$ rather than
$dx$ can be seen as the substitution of variable that makes the
integral tractable.

It is worth noting that, while James Bernoulli's investigation of
the mathematical curve describing the elastica is complete and sound, as a
more general work on the theory of elasticity there are some
problems. In particular, Bernoulli assumes (incorrectly) that the
inner curve of the elastica (AQR in Figure \ref{fig-elastica1694} is
the ``neutral fiber'', preserving its length as the elastica bends. In
fact, determining the neutral fiber is a rather tricky problem whose
general solution can still be determined only with numerical
techniques (see Truesdell \cite[pp. 96--109]{Truesdell60} for more
detail). Fortunately, in the ideal case where the thickness approaches
zero, the exact location of the neutral fiber is immaterial, and all
that matters is the relation between the moment and curvature.


\begin{figure*}
\begin{center}
\includegraphics[width=5in]{figs/huygens.png}
\caption{\label{huygens}Huygens's 1694 objection to Bernoulli's solution.}
\end{center}
\end{figure*}

The limitations of Bernoulli's approach were noted at the
time \cite{Fraser91}. Huygens published a short note in the \emph{Acta
  eruditorum} in 1694, very shortly after Bernoulli's publication in
the same forum, illustrating several of the possible shapes the
elastica might take on, and pointing out that Bernoulli's quadrature
only expressed the rectangular elastica. His accompanying figure is
reproduced here as Figure \ref{huygens}. The shapes are shown from
left to right in order of increasing force at the endpoints, and shape
A is clearly the rectangular elastica.

Bernoulli acknowledged this criticism (while pointing out that he
described these other cases explicitly in his paper, something that
Huygens apparently overlooked) and indicated that his technique could
be extended to handle these other cases (by using a non-zero constant
for the integration of Equation \ref{bernoulli-integral}), and went on
to give an equation with the general solution, reproduced by
Truesdell \cite[p. 101]{Truesdell60} as:

\begin{equation}
\label{bernoulli-general}
\pm dy = \frac{(x^2 \pm ab) dx}{\sqrt{a^4 - (x^2
    \pm ab)^2}}
\end{equation}

It is not clear that the publication of this more general solution had
much impact.  Bernoulli did not graph the other cases, nor did he seem
to be aware of other important properties of the solution, notably its
periodicity, nor the fact that it includes solutions with and without
inflections. (Another indication that these results were not generally
known is that Daniel Bernoulli, in a November 8, 1738 letter to
Leonhard Euler during a period of intense correspondence regarding the
elastica, wrote, ``Apart from this I have since noticed that the idea
of my uncle Mr. James Bernoulli includes all
elasticas.'' \cite[p.109]{Truesdell60})

Without question, the Bernoulli family set the
stage for Euler's definitive analysis of the elastica. The next
breakthrough came in 1742, when Daniel Bernoulli (the nephew of James)
proposed to Euler solving the general elastica problem with the the
technique that would finally crack it: variational analysis. This
general problem concerns the family of curves arising from an elastic
band of arbitrary given length, and arbitrary given tangent
constraints at the endpoints.

\section{Daniel Bernoulli proposes variational techniques---1742}

Daniel Bernoulli, in an October 1742 letter to
Euler \cite{BernoulliDan1742}, discussed the general problem of the
elastica, but had not yet managed to solve it himself:

\begin{quote}
Ich m\"ochte wissen ob Ew. die curvaturam laminae elasticae nicht
k\"onnten sub hac facie solviren, dass eine lamina datae longitudinis
in duobus punctis positione datis fixirt sey, also dass die tangentes
in istis punctis such positione datae seyen. ... Dieses ist die idea
generalissima elasticarum; hab aber sub hac facie noch keine Solution
gefunden, als per methodum isoperimetricorum, da ich annehme, dass die
vis viva potentialis laminae elasticae insita m\"usse minima seyn, wie
ich Ew. schon einmal gemeldet.
Auf diese Weise bekomme ich eine aequationem differentialem 4ti
ordinis, welche ich nicht hab genugsam reduciren k\"onnen, um zu
zeigen, dass die aequatio ordinaria elastica general sey.
\end{quote}

A rough translation into English reads:

\begin{quote}
I'd like to know whether you might not solve the curvature of the
elastic lamina under this condition, that on the length of the lamina
on two points the position is fixed, and that the tangents at these
points are given. ... This is the idea of the general elastica; I have
however not yet found a solution under this condition by the
isoperimetric method, given my assumption that the potential energy of
the elastic lamina must be minimal, as I've mentioned to you before.
In this way I get a 4th order differential equation, which I have not
been able to reduce enough to show a regular equation for the general
elastica.
\end{quote}

This previous mention is likely his 7 March 1739 letter to Euler,
where he gave a somewhat less elegant formulation of the potential
energy of an elastic lamina, and suggested the ``isoperimetric
method,'' an early name for the calculus of variations. Many founding
problems in the calculus of variations concerned finding curves of
fixed length (hence isoperimetric), minimizing or maximizing some
quantity such as area enclosed. Usually additional constraints are
imposed to make the problem more challenging, but, even in the
unconstrained case, though the answer (a circle) was known as early as
Pappus of Alexandria around 300 A.D, rigorous proof was a long time
coming.

In any case, Bernoulli ends the letter with what is likely the first
clear mathematical statement of the elastica as a variational problem
in terms of the stored energy\footnote{That said, James Bernoulli in
  1694 expressed the equivalence of the elastica with the lintearia,
  which has a variational formulation in terms of minimizing the
  center of gravity of a volume of water contained in a cloth sheet,
  as pointed out by Truesdell \cite[p. 201]{Truesdell60}}:

\begin{quote}
Ew. reflectiren ein wenig darauf, ob man nicht k\"onne, sine
interventu vectis, die curvaturam ABC immediate ex principiis
mechanicis deduciren.
Sonsten exprimire ich die vim vivam potentialem laminae elasticae
naturaliter rectae et incurvatae durch $\int\frac{ds}{RR}$, sumendo
elementum $ds$ pro constante et indicando radium osculi per $R$. Da
Niemand die methodum isoperimetricorum so weit perfectionnieret, als
Sie, werden Sie dieses problema, quo requiritur ut $\int\frac{ds}{RR}$
faciat minimum, gar leicht solviren.
\end{quote}

\begin{quote}
You reflect a bit on whether one cannot, without the intervention of
some lever, immediately deduce the curvature of ABC from the
principles of mechanics. Otherwise, I'd express the potential energy
of a curved elastic lamina (which is straight when in its natural
position) through $\int\frac{ds}{RR}$, assuming the element $ds$ is
constant and indicating the radius of curvature by $R$. There is
nobody as perfect as you for easily solving the problem of minimizing
$\int\frac{ds}{RR}$ using the isoperimetric method.
\end{quote}

Daniel was right. Armed with this insight, Euler was indeed able to
definitively solve the general problem within the year (in a letter
dated 4 September 1743, Daniel Bernoulli thanks Euler for mentioning
his energy-minimizing principle in the ``supplemento''), and this
solution was published in book form shortly
thereafter.\footnote{Truesdell also points out that this formulation wasn't
  entirely novel; Daniel Bernoulli and Euler had corresponded in 1738
  about the more general problem of minimizing $\int r^m\, ds$, and
  they seemed to be aware that the special case $m = -2$ corresponded
  to the elastica \cite[p. 202]{Truesdell60}. The progress of
  knowledge, seen as a grand sweep from far away, often moves in
  starts and fits when seen up close.}

%, and proposed an incredibly
%powerful technique for that analysis: the variational approach, in
%particular considering the curve which minimizes the value of some
%function.

\section{Euler's analysis---1744}
\label{elastica-euler}

Euler, building on (and crediting) the work of the Bernoullis, was the
first to completely characterize the family of curves known as the
elastica, and published this work as an appendix \cite{Euler1744} in
his landmark book on variational techniques. His treatment was quite
definitive, and holds up well even by modern standards.

Closely following Daniel Bernoulli's suggestion, he expressed the
problem of the elastica very clearly in variational form. He
wrote \cite[p. 247]{Euler1744}:

%2. Sit AB lamina Elastica utcunque incurvata; vocetur arcus AM = $s$,
%   \& radius osculi curv\ae\ MR = $R$: atque, secundum {\sc
%   Bernoullium,} exprimetur \emph{vis potentialis} in lamin\ae\
%   portione AM contenta hac formula $\int \frac{ds}{RR}$, siquidem
%   lamina sit ubique \ae qualiter crassa, lata \& elastica, atque in
%   statu naturali in directum extensa.

%...

\begin{quotation}
ut, inter omnes curvas ejusdem longitudinis, qu\ae\ non solum per
puncta A \& B transeant, sed etiam in his punctis a rectis positione
datis tangantur, definiatur ea in qua sit valor hujus expressionis
$\int \frac{ds}{RR}$ minimus.
\end{quotation}

In English:

\begin{quotation}
That among all curves of the same length that not only pass through
points A and B but are also tangent to given straight lines at these
points, it is defined as the one minimizing the value of the
expression $\int \frac{ds}{RR}$.
\end{quotation}

%\emph{todo: credit Alex Stepanov with translation help}

Here, $s$ refers to arclength, exactly as is common usage today, and
$R$ is the radius of curvature (``radius osculi curv\ae '' in Euler's
words), or $\kappa^{-1}$ in modern notation. Thus, today we are more
likely to write that the elastica is the curve minimizing the energy
$E[\kappa]$ over the length of the curve $0 \leq s \leq l$:

\begin{equation}
\label{elastica-def}
E[\kappa(s)] = \int_0^l \kappa(s)^2\ ds
\end{equation}

While today we would find it more convenient to work in terms of
curvature (intrinsic equations), Euler quickly moved to Cartesian
coordinates, using the standard definitions $ds = \sqrt{1 + y'^2}\,dx$
and $\frac{1}{R} = \frac{y''}{(1 + y'^2)^\frac{3}{2}}$, where $y'$ and
$y''$ represent $dy/dx$ and $d^2y/dx^2$, respectively. Thus, the
variational problem becomes finding a minimum for:

\begin{equation}
\label{euler-variation-quad}
\int \frac{y''^2}{(1 + y'^2)^{5/2}}\, dx
\end{equation}

This equation is written in terms of the first and second derivatives
of $y(x)$, and so the simple Euler-Lagrange equation does not
suffice. Daniel Bernoulli had run into this difficulty, as he
expressed in his 1742 letter. In hindsight, we now know that
expressing the problem in terms of tangent angle as a function of
arclength yields to a first-derivative varational approach, but this
apparently was not clear to either Bernoulli or Euler at the time.

In any case, by 1744, Euler had discovered what is now known as the
Euler-Poisson equation, capable of solving variational problems in
terms of second derivatives,
% I wish I knew whether the following parenthetical was true:
%(indeed, this was one of the great
%advances of the main body of his \emph{Method inveniendi})
and he
could apply it straightforwardly to Equation
\ref{euler-variation-quad}. He thus derived the following general
equation, where $a$ and $c$ are parameters (see Truesdell for a more
detailed explanation of Euler's
derivation \cite[pp. 203--204]{Truesdell60}):

%\[
%{dy \over dx} = {\alpha + \beta x + \gamma x^2 \over \sqrt{a^4 -
%    (\alpha + \beta x + \gamma x^2)^2}}
%\]

\begin{equation}
\label{euler-elastica-eq}
\frac{dy}{dx} = \frac{a^2 - c^2 + x^2}{\sqrt{(c^2 - x^2)(2a^2 - c^2 + x^2)}}
\end{equation}

This equation is essentially the same as James Bernoulli's general solution,
Equation \ref{bernoulli-general}.

Euler then goes on to classify the solutions to this equation based on
the parameters $a$ and $c$. To simplify constants, we propose the use
of a single $\lambda$ to replace both $a$ and $c$; this formulation
has a particularly simple interpretation as the Lagrange multiplier
for the straightforward variational solution of Equation \ref{elastica-def}.

\begin{equation}
\label{define-lambda}
\lambda = \frac{a^2}{2 c^2}
\end{equation}


Euler observes that there is an infinite variety of elastic curves,
but that ``it will be worth while to enumerate all the different kinds
included in this class of curves. For this way not only will the
character of these curves be more profoundly perceived, but also, in
any case whatsoever offered, it will be possible to decide from the
mere figure into what class the curve formed ought to be put. We shall
also list here the different kinds of curves in the same way in which
the kinds of algebraic curves included in a given order are commonly
enumerated.'' \cite[\S 14]{Euler1744}. According to the note in
Oldfather's translation,  Euler is referring to Newton's famous
classification of cubic curves \cite[p. 152, Note 5]{Oldfather33}. In
any case, Euler finds nine such classes, enumerated in the
table below:

\[
\begin{array}{rcccc}
\mbox{Euler's species \#} & \mbox {Euler's Figure} & \mbox{Euler's
  parameters} & \lambda & \mbox{comments} \\
1 &   & c = 0 & \lambda = \infty & \mbox{straight line} \\
2 & 6 & 0 < c < a & 0.5 < \lambda  \\
3 &   & c = a & \lambda = 0.5 & \mbox{rectangular elastica} \\
4 & 7 & a < c < a\sqrt{1.651868} & .302688 < \lambda < .5 \\
5 & 8 & c = a\sqrt{1.651868} & \lambda = .302688 & \mbox{lemnoid} \\
6 & 9 & a\sqrt{1.651868} < c < a\sqrt{2} & .25 < \lambda < .302688 \\
7 & 10 & c = a\sqrt{2} & \lambda = .25 & \mbox{syntractrix} \\
8 & 11 & a\sqrt{2} < c & 0 < \lambda < .25 \\
9 & & a = 0 & \lambda = 0 & \mbox{circle}
\end{array}
\]

\begin{figure*}
\begin{center}
\includegraphics[scale=.7]{figs/0334.png}
\caption{\label{euler-tab-iii}Euler's elastica figures, Tabula III.}
\end{center}
\end{figure*}

\begin{figure*}
\begin{center}
\includegraphics[scale=.7]{figs/0336.png}
\caption{\label{euler-tab-iv}Euler's elastica figures, Tabula IV.}
\end{center}
\end{figure*}

Euler includes figures for six of the nine cases, reproduced here in
Figures \ref{euler-tab-iii} and \ref{euler-tab-iv}. Of the three
remaining, species \#1 is a degenerate straight line, and species \#9
is a circle. Species \#3, the \emph{rectangular elastica,} is of
special interest, so it is rather disappointing that Euler did not
include a figure for it. Possibly, it was considered already known due
to Bernoulli's publication. Note also that in this special case, $c =
a$, the equation becomes equivalent to Bernoulli's Equation
\ref{bernoulli-elastica}.

Compare these figures with the computer-drawn Figure
\ref{elastica-fam}. It is remarkable how clearly Euler was able to
visualize these curves, even 250 years ago. It is likely that the
distortions and inaccuracies are due primarily to the draftsman
engraving the figures for the book rather than to Euler himself; they
display lack of symmetry that Euler clearly would have known. In fact,
Euler computed a number of values to seven or more decimal places,
including the values of $a$ and $c$ for the lemnoid shape (species \#5).

\begin{figure*}
\begin{center}
\includegraphics[scale=.7]{figs/elastica-fam}
\caption{\label{elastica-fam}The family of elastica solutions.}
\end{center}
\end{figure*}

Note that for $c^2 < 2a^2$, or $\lambda > .25$, the inflectional cases,
the equation is well-defined for $-c < x < c$. In these cases, the
parameter $c$ is half the width of the figure. In the non-inflectional
cases ($\lambda < .25$), the equation has no solutions at $x = 0$, and
in Euler's figure the curve is drawn to the right of the $y$ axis.

The curve described by species \#7, the only nonperiodic solution, was
known to Poleni in 1729, and is also known as the ``syntractrix of
Poleni,'' or, in French, ``la courbe des for\c{c}ats'' (the curve of
convicts, or galley slaves). Euler gives its equation (in section 31)
in closed form (without citing Poleni):

\begin{equation}
y = \sqrt{c^2 - x^2} - \frac{c}{2}\log \frac{c + \sqrt{c^2 - x^2}}{x}
\end{equation}

\subsection{Moments}

Euler was also clearly aware of the simple moment approach to the
elastica, and in section 43 of the Additamentum, demonstrated its
equivalence to the variational approach. To do so, he manipulated the
quadrature formulation of the elastica, Equation
\ref{euler-elastica-eq}, grouping all the terms involving first and
second derivatives of the curve into a single term
$Sq(1-p^2)^{-\frac{3}{2}}$. Recall that $p = \frac{dy}{dx}$ and $q =
\frac{dp}{dx} = \frac{d^2y}{dx^2}$. Thus, this term is $S$ times the
curvature, yielding an equation relating curvature to Cartesian
coordinates. Euler explains:

\begin{quote}
``But $\frac{-(1 +
pp)^{3 : 2}}{q}$ is the radius of curvature R; whence, by doubling the
  constants $\beta$ and $\gamma$, the following equation will arise:

\begin{equation}
\label{euler-moment-eq}
\frac{S}{R} = \alpha + \beta x - \gamma y\,.
\end{equation}

This equation agrees admirably with that which the second or direct
method supplies. For let $\alpha + \beta x - \gamma y$ express the
moment of the bending power, taking any line you please as an axis, to
which moment the absolute elasticity $S$, divided by the radius of
curvature $R$ must be absolutely equal. Thus, therefore, not only has
the character of the elastic curve observed by the celebrated
{\sc Bernoulli} been most abundantly demonstrated, but also the very
great utility of my somewhat difficult formulas has been established
in this example.''
\end{quote}

This result certainly is confirmation of the variational technique,
but in fairness it must be pointed out that James Bernoulli was able
to derive essentially the identical equation (again, note the
similarity between Equations \ref{bernoulli-general} and
\ref{euler-elastica-eq}) by using a combination of mechanical insight
and clever integration.

%Note that this moment-based formulation covers the general case, not
%just the rectangular elastica studied by James Bernoulli. However, it
%took Euler's full abilities in the calculus of variations to fully
%solve the elastica problem til he could realize this fact.

\section{Elliptic integrals}

The elastica, having been present at the birth of the variational
calculus, also played a major role in the development of another
branch of mathematics: the theory of elliptic functions.

Even as the quadratures of these simple curves came to be revealed,
analytic formulae for their lengths remained elusive. The functions
known by the first half of the 18th century were insufficient to
determine the length even of a curve as well-understood as an ellipse.

James Bernoulli set himself to this problem and was able to pose it
succinctly (and even compute approximate numerical values), if not
fully solve it himself \cite{Sridharan04}.  If the quadrature of a
unit-excursion rectangular elastica is given by this equation:

\begin{equation}
y = \int_0^x \frac{x^2dx}{\sqrt{1-x^4}},
\end{equation}

then the arc length is given by:

\begin{equation}
\label{lemniscate-integral}
s = \int_0^x \frac{dx}{\sqrt{1-x^4}}.
\end{equation}

\begin{figure*}
\begin{center}
\includegraphics[scale=0.6]{figs/lemni.pdf}
\caption{\label{lemni}Bernoulli's lemniscate.}
\end{center}
\end{figure*}

This integral is now called the ``Lemniscate integral'', because of
its connection with the lemniscate, another beautiful curve studied by
James Bernoulli (Figure \ref{lemni}). The length of the lemniscate is
equal to that of the rectangular elastica; while Equation
\ref{lemniscate-integral} gives the arclength of the elastica as a
function of the $x$ coordinate, this nearly identical
equation relates arclength to the radial coordinate $r$ in the lemniscate:

\begin{equation}
\label{lemniscate-integral-rad}
s = \int_0^r \frac{dr}{\sqrt{1-r^4}}.
\end{equation}

%The lemniscate's arc length $s$
%as a function of $x$ is governed by the same equation. The same
%equation can generate both curves due to different starting
%angles---in the case of the elastica, it starts in the direction of
%increasing x, and in the lemniscate, it is perpendicular.

Among the lemniscate's other representations, its implicit equation in
Cartesian coordinates is a simple polynomial (no such
corresponding formulation exists for the elastica):

\begin{equation}
\label{lemniscate}
(x^2 + y^2)^2 = x^2 - y^2.
\end{equation}

Bernoulli approximated the integral for the arclength of the
lemniscate using a series expansion and determined upper and lower
numerical bounds, but felt the calculation of them would not fall to
standard analytical techniques. He wrote, ``I have heavy grounds to
believe that the construction of our curve depends neither on the
quadrature nor on the rectification of any conic section.''
\cite{Truesdell83}. Here, ``quadrature'' means the area under the
curve, or the indefinite integral, and ``rectification'' means the
computation of the length of the curve. Bernoulli's prediction would
not turn out to be entirely accurate; as we shall see, the curve would
later be expressed in terms of Jacobi elliptic functions, which in
turn are deeply related to the question of determining the arclength
(rectification) of the ellipse.

Fagnano took up the problem of finding the length of the lemniscate,
and achieved some impressive results. Indeed, Jacobi fixes the date
for the birth of elliptic functions as 23 December 1751, when Euler
was asked to review Fagnano's collected works. However, Euler had
begun study of elliptic integrals as early as 1738, when he wrote to
the Bernoullis that he had ``noticed a singular property of the
rectangular elastica'' having unit excursion \cite{Truesdell83}:

\begin{equation}
\mbox{length} \cdot \mbox{height} = \int_0^1 \frac{dx}{\sqrt{1-x^4}}
\cdot
\int_0^1 \frac{x^2dx}{\sqrt{1-x^4}} = \frac{1}{4}\pi.
\end{equation}

After receiving Fagnano's work, Euler caught fire and started his
remarkable research, first on lemniscate integrals, then on the more
general problem of elliptic integrals, especially the discovery of
the addition theorems for elliptic functions in the 1770s. The reader
interested in more details, including mathematical derivations, is
directed to Sridharan's delightful historical sketch \cite{Sridharan04}.

These integrals would ultimately be the basis for  closed-form solutions of the
elastica equation, with both curvature and Cartesian coordinates given
as ``special functions'' of the arclength parameter, as will be
described in Section \ref{jacobi-section}.

\section{Laplace on the capillary---1807}

Remarkably, the elastica appears as yet another shape of the solution
of a fundamental physics problem---the capillary. Pierre Simon Laplace
investigated the equation for the shape of the capillary in his 1807
\emph{Suppl\'ement au dixi\`eme livre du Trait\'e de m\'ecanique
  c\'eleste. Sur l'action capillaire}\footnote{reprinted in Volume 4 of
Laplace's Complete Works \cite[pp. 349--401]{LaplaceWorks4}}. See I. Grattan Guinness
\cite[p. 442]{Grattan90} for a thorough description of these
results. In particular, Laplace considers the surface of a fluid trapped
between two vertical plates, and obtains this equation (722.16 in
\cite{Grattan90}):

\begin{equation}
z'' = 2(\alpha z + b^{-1})(1 + z'^2)^{3/2}
\end{equation}

Here, $z$ represents height, and $z'$ and $z''$ represent first and
second derivatives with respect to the horizontal coordinate. With
suitable renaming of coordinates and substution of
Equation \ref{newton-curvature-eq} for curvature in Cartesian coordinates,
this equation can readily seen to be equivalent to Euler's Equation
\ref{euler-moment-eq}.

Laplace also recognized that at least one instance of his equation was
equivalent to the elastica. He derives Equation
\ref{bernoulli-integrated} (save that Laplace writes Z for S) and
notes that it is identical to the elastic curve
\cite[p. 379]{LaplaceWorks4}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=2.7in]{figs/maxwellfig6.pdf}
\includegraphics[width=2in]{figs/maxwellfig8.pdf}
\caption{\label{capillary-fig}Maxwell's figures for capillary action.}
\end{center}
\end{figure}


It is not clear when the general equivalence between the capillary
surface and the elastica was first appreciated. The special case of a
single plate is published in \emph{The Elements of Hydrostatics and
  Hydrodynamics} \cite[p. 32]{Miller1831} in 1833. In any case, by
James Clerk Maxwell was fully aware of it, including it his article
``Capillary Action'' in the 9th edition of the Encyclop\ae dia
Britannica \cite{Maxwell1890}, as well as the result that the
cross-section of the capillary surface in a cylindrical tube is also
an elastica. His figures (reproduced here as Figure
\ref{capillary-fig}) illustrate the problem and clearly show a
non-inflectional, periodic instance of the elastica.

\section{Kirchhoff's kinetic analogy---1859}

Surprisingly, the differential equation for the elastica, expressing
curvature as a function of arclength, are equivalent to those of the
motion of the pendulum, as worked out by Kirchhoff in 1859 (see
\cite{DAntonio07} for historical detail). Using simple variational
techniques, taking arclength as the independent variable and angle
$\theta$ from the horizontal coordinate as the dependent, Equation
\ref{elastica-def} yields (the sine and cosine arise from the
specification of endpoint location constraints, which are described as
an integral of the sine and cosine of $\theta$ over the length of the
curve):

\begin{equation}
\label{elastica-eq}
\theta'' + \lambda_1 \sin \theta
+ \lambda_2 \cos \theta = 0
\end{equation}

This differential equation for the shape of the
elastica is mathematically equivalent to that of the dynamics of a
simple swinging pendulum. This kinetic analog is useful for developing
intuition about the solutions of the elastica equation. In particular,
it's easy to see that all solutions are periodic, and that the family
of solutions is characterized by a single parameter (modulo scaling,
translation, and rotation of the system).

\begin{figure*}
\begin{center}
\includegraphics[scale=.6]{figs/pendulum}
\caption{\label{pendulum}An oscillating pendulum.}
\end{center}
\end{figure*}

Consider, as shown in Figure \ref{pendulum}, a weight of mass $m$ at
the end of a pendulum of length $r$, with angle from vertical
specified as a function of time $\theta(t)$.

Then the velocity of the mass is $r\theta'$ (where, in this section,
the $'$ notation represents derivative with respect to time), and its
acceleration is $r\theta''$. The net force of gravity, acting with the
constraint of the pendulum's angle is $mg \sin \theta$, and thus we
have:

\begin{equation}
F_{net} = ma = mr\theta'' = mg \sin \theta
\end{equation}

With the substitution $\lambda_1 = -g/r$, and the replacement of
arclength $s$ for time $t$, this equation becomes equivalent to
Equation \ref{elastica-eq}, the equation of the elastica. Note that
angle (as a function of arclength) in the elastica corresponds to
angle (as a function of time) in the pendulum, and the elastica's
curvature corresponds to the pendulum's angular momentum.

% According to Love\cite{Love}, the systematic application of the
%kinetic analogue was worked out by W. Hess in 1885.

The swinging of a pendulum is perhaps the best-known example of a
periodic system. Further, it is intuitively easy to grasp the
parameter space of the system. Transformations of scaling (varying the
pendulum length) and translation (assigning different phases of the
pendulum swing to $t = 0$) leave the solutions essentially unchanged.
Only one parameter, how high the pendulum swings, changes the
fundamental nature of the solution.

The solutions of the motion of the pendulum, as do the solutions of
the elastica Equation \ref{elastica-eq}, form a family characterized
by a single scalar parameter, once the trivial transforms of rotation
and scaling are factored out. Without loss of generality, let $t = 0$
at the bottom of the swing (i.e. maximum velocity) and let the
pendulum have unit length. The remaining parameter is then the ratio
of the total energy of the system (which remains unchanged over time)
to the potential energy of the pendulum at the top of the swing (the
maximum possible), in both cases counting the potential energy at the
bottom of the swing as zero. Thus, the total system energy is also
equal to the kinetic energy at the bottom of the swing.

For mathematical convenience, define the parameter $\lambda$ as one
fourth the top-of-swing potential energy divided by the total energy
(the constant is chosen so that $\lambda$ matches the Lagrange multiplier
$\lambda_1$ in Equation \ref{elastica-eq}). This convention is
equivalent to varying the force of gravity while keeping the maximum
kinetic energy fixed, which may be justified by noting that the
zero-gravity case (which would require infinite kinetic energy if we
were assuming unit gravity) is far more relevant in the context of
splines (it corresponds, after all, to a circular arc, so any
elastica-based spline meeting the criterion of roundness must
certainly have this solution) than the converse of zero kinetic
energy.

The potential energy at the top of the pendulum is $2mgr$. The kinetic
energy at the bottom of the swing is $\frac{1}{2} mv^2$, where, as
above, $v = r\theta'$. Thus, according to the definition above,

\[
\lambda = \frac{1}{4}\frac{2mgr}{\frac{1}{2m(r\theta')^2}} =
\frac{g}{r(\theta')^2}
\]

In other words, assuming (without loss of generality) unit pendulum
length and unit velocity the bottom of the swing, $\lambda$ simply
represents the force of gravity. And this $\lambda$ is the same as
defined in Equation \ref{define-lambda}.

%Unfortunately, while the kinetic analogy generated a fair amount of
%excitement, it does not generalize in any meaningful way to other
%results. The fact that the same equation governs the behavior of two
%disparate systems is probably more a testament to the simplicity than
%anything else.

\section{Closed form solution: Jacobi elliptic functions---1880}
\label{jacobi-section}

The closed-form solutions of the elastica, worked out by Saalsch\"utz
in 1880 \cite{Saalschutz1880}, rely heavily on Jacobi elliptic
functions (see also \cite{DAntonio07} for more historical development,
and Greenhill \cite{Greenhill1892} for a contemporary presentation of
these results in English, suggesting the kinetic analog was a major
motivator to deriving the closed form solutions).

The \emph{Jacobi amplitude} $\mbox{am}(u, k)$ is defined as the
inverse of the Jacobi elliptic integral of the first kind:

\begin{equation}
\begin{gathered}
\mbox{am}(u, m) = \mbox{the value of $\phi$ such that}\\
u = \int_0^\phi\!\! \frac{1}{\sqrt{1 - m \sin^2 t}}dt
\end{gathered}
\end{equation}

Here, $m$ is known as the \emph{parameter}. Jacobi elliptic functions
are also commonly written in terms of the \emph{elliptic modulus} $k =
\sqrt m$.

From this amplitude, the doubly periodic generalizations of the
trigonometric functions follow:

\begin{gather}
\mbox{sn}(u, m) = \sin (\mbox{am} (u, m) ) \\
\mbox{cn}(u, m) = \cos (\mbox{am} (u, m) ) \\
\mbox{dn}(u, m) = \sqrt {1 - m \sin^2 (\mbox{am} (u, m))}
\end{gather}

Note that, when $m$ is zero, sn and cn are equivalent to sin and cos,
respectively, and when $m$ is unity, sn and cn are equivalent to tanh
and sech ($1/\mbox{cosh}$)(\cite[p. 571]{abramowitz+stegun}, \S 16.6).

\subsection{Inflectional solutions}

The closed form solutions are given separately for inflectional and
non-inflectional cases. The intrinsic form, curvature as a function of
arclength, is fairly simple:

\begin{equation}
\frac{d \theta}{ds} = \kappa = 2\sqrt m\ \mbox{cn}(s, m),
\end{equation}

Expressing the curve in the form of angle as a function of
arclength is also straightforward:

\begin{equation}
\sin \frac{1}{2} \theta = \sqrt m\ \mbox{sn}(s, m)
\end{equation}

Through an impressive feat of analysis, the curve can be expressed in
cartesian coordinates as a function of arclength. First, we'll need
the elliptic integral of the second kind, $E(\phi, k)$:

\begin{equation}
\label{E-def}
E(\phi, k) = \int_0^\phi \sqrt{1-m\sin^2\theta}\; d\theta
\end{equation}

\begin{equation}
\begin{array}{l}
x(s) = s - 2E(\mbox{am}(s, m), m) \\
y(s) = -2\sqrt{m}\; cn(s, m)
\end{array}
\end{equation}

In Love's presentation of these results, $E(\mbox{am}(s, m), m)$ is
defined in terms of an integral of $\mbox{dn}(u, m)$. The equivalence
with Equation \ref{E-def} follows through a standard identity of the
Jacobi elliptical functions (\cite[p. 576]{abramowitz+stegun}, 16.26.3).

\begin{equation}
E(\mbox{am}(s, m), m) = \int_0^s (\mbox{dn}(u, m))^2\; du
\end{equation}

\begin{figure}
\begin{center}
\includegraphics[width=3in]{figs/roulette.pdf}
\caption{\label{roulette-fig}Rectangular elastica as roulette of hyperbola.}
\end{center}
\end{figure}

The solution in terms of elliptic integrals opened the way to a few
more curious results. For one, the \emph{roulette} of the center of a
rectangular hyperbola is a rectangular
elastica \cite{Greenhill1892}. More precisely, let the hyperbola roll
along a line without slipping. Then, the curve traced by its center is
a rectangular elastica, as shown in Figure \ref{roulette-fig}. The general term for a roulette formed from a
conic section is a Sturm's roulette.\footnote{Visit
  http://www.mathcurve.com/courbes2d/sturm/sturm.shtml for an animated
demonstration of this property.}

\subsection{Non-inflectional solutions}

The equations in the non-inflectional case are similar.

\begin{equation}
\frac{d\theta}{ds} = \kappa = \frac{2}{\sqrt{m}}\; \mbox{dn}\Big
(\frac{s}{\sqrt{m}}, m\Big )
\end{equation}

\begin{equation}
\sin \frac{1}{2} \theta = \mbox{sn}\Big (\frac{s}{\sqrt{m}}, m\Big )
\end{equation}

For the Cartesian version, and for a more detailed derivation, refer
to Love \cite{Love}.

Truesdell considers the popularity of the elliptic function approach
to be a mixed blessing for mechanics and for mathematics. Where
Euler's method used direct nonlinear thinking based on physical
principles, much of the elliptic literature explored properties of
special functions, which ``came to be ends of research rather than
means to solve a natural problem.'' \cite{Truesdell83} Indeed, elliptic
functions are rarely used today for computation of elastica, in favor
of numerical methods. Indeed, I've used a simple 4th-order Runge Kutte
differential equation solver to draw the figures for this work, due to
its good convergence and efficiency and simple expression in code. One
particularly unappealing aspect of the elliptic approach is the sharp
split between inflectional and non-inflectional cases, while one
differential equation smoothly covers both cases.

Even so, Jacobi elliptic functions are now part of the mainstream of
special functions, and fast algorithms for computing them are
well-known \cite{NR}. Elliptic functions are still the method of choice
for the fastest computation of the shape of an elastica.

\section{Max Born---1906}

In spite of the equation for the general elastica being published as
early as 1695, the curves had not been accurately plotted until Max
Born's 1906 PhD thesis, ``Investigations of the
stability of the elastic line in the plane and in space'' \cite{Born1906}.

\begin{figure}
\begin{center}
\includegraphics[angle=270,width=3in]{figs/born.pdf}
\caption{\label{born}Born's experimental apparatus for measuring the elastica.}
\end{center}
\end{figure}

Born also constructed experimental apparatus using weights and dials
to place the elastic band in different endpoint conditions, and used
photographs to compare the equations to actual shapes. An example
setup is shown in Figure \ref{born}, which shows an inflectional
elastica under fairly high tension, $\lambda \approx 0.26$.

Years later, Born wrote, ``...I felt for the first time the delight of
finding a theory in agreement with measurement---one of the most
enjoyable experiences I know.'' \cite[p. 21]{Born68}. Born used the
best modern mathematical techniques to address the problem of the
elastica, and, among other things, was able to generalize it to the
three dimensional case of a wire in space, not confined to the plane.

\section{Influence on modern spline theory---1946 to 1965}

Mechanical splines made of wood or metal have long been an inspiration
for the mathematical concept of spline (and for its
name). Schoenberg's justification for cubic splines in 1946 was a
direct appeal to the notion of an elastic strip. Schoenberg's main
contribution was to define his spline in terms of a variational
problem closely approximating Equation \ref{elastica-def}, but making
the small-deflection approximation.
\footnote{Note: what follows is the
  corrected version as appears in his selected
  papers \cite{SchoenbergPapers}.}

\begin{quote}

\textbf{3.1 Polynomial spline curves of order $k$.} A spline is a
simple mechanical device for drawing smooth curves. It is a slender
flexible bar made of wood or some other elastic material. The spline
is placed on the sheet of graph paper and held in place at various
points by means of certain heavy objects (called ``dogs'' or ``rats'')
such as to take the shape of the curve we wish to draw. Let us assume
that the spline is so placed and supported as to take the shape of a
curve which is nearly parallel to the $x$-axis. If we denote by $y =
y(x)$ the equation of this curve then we may neglect its small slope
$y'$, whereby its curvature becomes

\[ 1/R = y'' / (1 + y'^2)^{3/2} \approx y''
\]

The elementary theory of the beam will then show that the curve $y =
y(x)$ is a polygonal line composed of cubic arcs which join
continuously, with a continuous first and second
derivative\footnote{Schoenberg is indebted for this suggestion to
Professor L. H. Thomas of Ohio State University.}. These junction
points are precisely the points where the heavy supporting objects are
placed.

\end{quote}

While Schoenberg's splines are excellent for fitting the values of
functions (the problem of appoximation), it was clear that they were
not ideal for representing shapes. Birkhoff and de Boor wrote in
1965 \cite{Birkhoff65} that ``linearized interpolation schemes have a
basic shortcoming: they are \emph{not intrinsic} geometrically because
they are not invariant under rigid rotation. Physically it seems more
natural to replace linearized spline curves by \emph{non-linear}
splines (or ``elastica''), well known among elasticians,'' citing
Love \cite{Love} as an authority. They also stated the result that in a
mechanical spline constrained to pass through the control points only
by pure \emph{normal forces}, only the rectangular elastica is needed.

Birkhoff and de Boor also noted some shortcomings in the elastica as a
replacement for the mechanical spline, particularly the lack of an
existence and uniqueness theory for non-linear spline curves with
given endpoints, endslopes, and sequence of internal points. Indeed,
they state that ``nor does it seem particularly desirable'' to have
techniques for intrinsic splines approximate the elastica, and they
proposed other techniques, such as Hermite interpolation by segments
of Euler spirals joined together with continuous curvature. 

According to notes taken by Stanford professor George
Forsythe \cite{Forsythe1964}, in the conference presentation of this
work, they also referenced Max Born's PhD thesis \cite{Born1906}, and
also described a goal of the work as providing an ``automatic French
curve.'' The presentation must have made an impression on Forsythe,
judging from the laconic sentence, ``Deep.'' Forsythe, working with his
colleague Erastus Lee from the Mechanical Engineering college at
Stanford, would go on to analyze the nonlinear spline considerably
more deeply, deriving it from both the variational formulation of
Equation \ref{elastica-def} and an exploration of moments, normal
forces, and longitudinal forces \cite{ForsytheLee}. That publication
also included a result for energy minimization in the case where
the elastica is a closed loop.

\section{Numerical techniques---1958 through today}

The arrival of the high-speed digital computer created a strong demand
for efficient algorithms to \emph{compute} the elastica, particularly
to compute the shape of an idealized spline constrained to pass
through a sequence of \emph{control points}.

Birkhoff and de Boor \cite{Birkhoff65} cited several approximations to
non-linear splines, including those of Fowler and
Wilson \cite{Fowler62} and MacLaren \cite{MacLaren58} at Boeing in 1959,
and described their own work along similar lines. None of these was a
particularly precise or efficient approximation.

Around the same time, and very possibly working independently, Mehlum
and others designed the Autokon system \cite{Mehlum74} using an
approximation to the elastica (called the {\sc kurgla 1}
algorithm). However, this algorithm was not an accurate simulation of
the mechanical spline, as did not minimize the total bending energy of
the curve, and, in fact, could generate (approximations to) the entire
family of elastica curves in service of its splines. An objection to
this technique is that the spline was not \emph{extensional;} adding a
new control point conciding with the generated spline would slightly
perturb the curve. Thus, this approximate elastica was soon replaced by
the {\sc kurgla 2} algorithm implementing precisely the Euler spiral
Hermite interpolation suggested by Birkhoff and de Boor.

A sequence of papers refining the numerical techniques for computing
the elastica-based nonlinear spline followed: Glass in
1966 \cite{Glass66}, Woodford in 1969 \cite{Woodford1969}, Malcolm in
1977 \cite{Malcolm1977}. Most of these involve discretized formulations
of the problem. Interestingly, Malcolm reports that Larkin developed a
technique based on direct evaluation of the Jacobi elliptic integrals
which is nonetheless ``probably quite slow'' compared with the
discretized approaches, ``due to the large number of transcedental
functions which must be evaluated.'' Malcolm's paper contains a good
survey of known numerical techniques, and is recommended to the reader
interested in following this development in more detail.

In spite of the fairly rich literature available on the elastica, at
least one researcher, B. K. P. Horn in 1981, seems to have
independently derived the rectangular elastica from the principle of
minimizing the strain energy \cite{Horn}, going through an impressive
series of derivations, and using the full power of elliptic integral
theory, to arrive at exactly the same integral as Bernoulli had
derived almost three hundred years previously.

Edwards \cite{Edwards92} proposes a spline solution in which the
elastica in the individual segments are computed using Jacobi
elliptic functions. He also gives a global solver based on a
Newton technique, each iteration of which solves a band-diagonal
Jacobian matrix. A particular concern was exploring the cases when no
stable solution exists, a weakness of the elastica-based spline
compared to other techniques. Edwards claims that his numerical
techniques would always converge on a solution if one exists.

\section{Sources and acknowledgements}

This historical sketch draws heavily on Truesdell's history in an
introduction to Ser 2, Volume X of Euler's \emph{Opera
Omnia} \cite{Truesdell60}. Indeed, a careful reading of that work
reveals virtually all that needs to be known about the problem of the
elastica and its historical development through the time of Euler.

Euler's 1744 \emph{Additamentum} is a truly remarkable work, deserving
of deep study. The original Latin is available in facsimile thanks to
the Euler archive, but the 1933 English translation by Oldfather
\cite{Oldfather33} makes it much more understandable to the
English-speaking reader.
 
I am deeply grateful to Alex Stepanov, Seth Schoen, Martin Meijering
and Ben Fortson for help with the translations from the Latin. Special
mention is also due the excellent ``Latin words'' program by William
Whitaker, which lets anyone with a rudimentary knowledge of Latin and
a good deal of patience and determination puzzle out the meaning of a
text. Of course, all errors in translation are my own responsibility.

Patricia Radelet has been most helpful in supplying high-quality
images for James Bernoulli's original figures, reproduced here as
Figures \ref{bernoulli-clxx} and \ref{fig-elastica1694}, and for
providing scans of the original text of his 1694 publication.
 
Thanks also to the Stanford Special Collections Library for giving me
access to Prof. Forsythe's notes and correspondence, which were
invaluable in tracing the influence of the elastica through early work
on non-linear splines.

\bibliographystyle{plain}
\bibliography{curves}

\end{document}
