Variations of Curves

Abstract

Many important special curves $\gamma$ arise by minimizing a certain variational energy $E(\gamma )$.

Many important special curves $\gamma$ arise by minimizing a certain variational energy $E(\gamma )$. For example, $E(\gamma )$ could be a linear combination of length and bending energy, in which case the curve is called an elastic curve. We are not only interested in minima but also in unstable energetic equilibria, possibly under constraints like fixing the curve near its end points. In this chapter we develop the basics of Variational Calculus. In particular, this allows us to explore elastic curves. Beyond straight lines and circles, these are the most important special curves in $\mathbb {R}^n$.

1 One-Parameter Families of Curves

On many occasions we will have to deal not only with individual curves $\gamma \colon [a,b]\to \mathbb {R}^n$ but with whole one-parameter families $t\mapsto \gamma _t$ of curves.

Definition 2.1

Let $g_t\colon [a,b]\to \mathbb {R}^k$ be a smooth map, defined for each $t\in [t_0,t_1]$. Then the one-parameter family of maps$[t_0,t_1]\ni t\mapsto g_t$ is called smooth if the map

$$\displaystyle \begin{aligned} [a,b]\times [t_0,t_1]&\to \mathbb{R}^k,\ (x,t)\mapsto g_t(x)\end{aligned} $$

is smooth (as always, in the sense of Remark 1.2).

Given a smooth one-parameter family

$$\displaystyle \begin{aligned} t\mapsto (g_t\colon[a,b]\to \mathbb{R}^k),\quad t\in [t_0,t_1]\end{aligned}$$

of maps, also

$$\displaystyle \begin{aligned} t\mapsto g_t^{\prime}\end{aligned}$$

is a smooth one-parameter family of maps $g_t^{\prime }\colon [a,b]\to \mathbb {R}^k$. The same holds for ) where ) is defined as

The dot and prime derivatives are just partial derivatives, so they commute by Schwarz’s theorem:

Theorem 2.2

For a smooth one-parameter family of maps$t\mapsto g_t$, where$g_t\colon [a,b]\to \mathbb {R}^k$we have

In our context, one-parameter families of maps will mainly arise as variations of a single map $g\colon [a,b]\to \mathbb {R}^k$:

Definition 2.3

A smooth one-parameter family $t\mapsto g_t$ of maps from M to $\mathbb {R}^k$ is called a variationof a smooth map$g\colon M\to \mathbb {R}^k$ if $t_0 < 0 < t_1$ and $g_0=g.$ In this context, we will also use the notation

Our main interest is in variations of curves $\gamma \colon [a,b]\to \mathbb {R}^n$ (and the associated variations of derived quantities like the unit tangent or the length):

Definition 2.4

For a variation $t\mapsto \gamma _t$ of a curve $\gamma \colon [a,b]\to \mathbb {R}^n$ the map

is called its variational vector field.

Suppose we have a smooth one-parameter-family $t\mapsto \gamma _t$ of curves $\gamma _t\colon [a,b]\to \mathbb {R}^n$, meaning that $\gamma _t^{\prime }(x)\neq 0$ for all $x\in [a,b]$ and all $t\in [t_0,t_1]$. Then we can think of this family (just for the purpose of intuition, no need for further formal definitions) as a smooth map from $[t_0,t_1]$ into in the space $\mathcal {M}$ of all curves $\gamma \colon [a,b]\to \mathbb {R}^n$. The vector ) can then be thought of as the “velocity vector” of that map at time t (see Fig. 2.1).

Fig. 2.1

A variation of a curve $\gamma$ can be interpreted as a map into the space $\mathcal {M}$ of all curves $\gamma \colon [a,b]\to \mathbb {R}^n$

Remark 2.5

Throughout this whole book we will treat $C^\infty \left ([a,b],\mathbb {R}^k\right )$ (and its analog in the context of surfaces) only as a vector space, based on notions from Linear Algebra. So, for example, we will indeed use the Euclidean inner product

$$\displaystyle \begin{aligned} \langle \! \langle g,h\rangle \!\rangle :=\int_a^b \langle g,h\rangle\end{aligned}$$

but we will never put any topology on $C^\infty ([a,b],\mathbb {R}^k)$. This means that you will get confused if you try to interpret what we say based on notions from Functional Analysis. These notions have important applications in Differential Geometry, but they are not used at all in this book.

2 Variation of Length and Bending Energy

Given a variation $t\mapsto \gamma _t$ of a curve $\gamma \colon [a,b]\to \mathbb {R}^n$, we want to determine $\left .\frac {d}{dt}\right |{ }_{t=0}\mathcal {L}(\gamma _t)$ and $\left .\frac {d}{dt}\right |{ }_{t=0}\mathcal {B}(\gamma _t)$. We first compute the time derivative of the integrands of these integrals:

Theorem 2.6

Let$t\mapsto \gamma _t$be a variation with variational vector field$Y\colon [a,b]\to \mathbb {R}^n$of a curve$\gamma \colon [a,b]\to \mathbb {R}^n$with speed$v=ds$and unit tangent field T. Then the variation of ds is given by

Proof

Differentiating the equation $v_t=|\gamma _t^{\prime }|$ with respect to t at $t=0$ we obtain

□

Before we proceed to compute the rate of change for the bending energy integrand, note that (unlike the situation for partial derivatives), for a one-parameter family $t\mapsto \gamma _t$ the derivative with respect to t does not commute with the derivative with respect to arclength:

Theorem 2.7

Let$t\mapsto \gamma _t$be a variation with variational vector field$Y\colon [a,b]\to \mathbb {R}^n$of a curve$\gamma \colon [a,b]\to \mathbb {R}^n$with speed$v=ds$. Then for any one-parameter family$t\mapsto g_t$of functions$g_t\colon [a,b]\to \mathbb {R}^k$with$g_0=:g$we have

Proof

By Theorem 2.6,

□

Theorem 2.8

Given a variation$t\mapsto \gamma _t$with variational vector field$Y\colon [a,b]\to \mathbb {R}^n$of a curve$\gamma \colon [a,b]\to \mathbb {R}^n$with speed$v=ds$, the corresponding variation of the bending energy density is

$$\displaystyle \begin{aligned} \left(\frac{1}{2}\left\langle \frac{dT}{ds}, \frac{dT}{ds}\right\rangle ds\right)^{\mathit{\mbox{\bf \hspace{-0.35ex}.}}} = \left(\left\langle\frac{d^2Y}{ds^2},\frac{dT}{ds}\right\rangle -\frac{3}{2}\left\langle \frac{dY}{ds},T\right\rangle\left\langle \frac{dT}{ds},\frac{dT}{ds}\right\rangle\right)ds.\end{aligned}$$

Proof

Applying Theorem 2.7 to $g=\gamma$ we obtain

Using this, Theorem 2.6, the fact that $\langle T,T\rangle =1$ implies $\left \langle \frac {dT}{ds},T\right \rangle =0$ and Theorem 2.7 with $g=T$ we obtain

□

The proof of the following theorem is based on applying integration by parts repeatedly.

Theorem 2.9

Given a variation$t\mapsto \gamma _t$with variational vector field$Y\colon [a,b]\to \mathbb {R}^n$of a curve$\gamma \colon [a,b]\to \mathbb {R}^n$, the corresponding variations of the length and bending energy are

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0}\mathcal{L}(\gamma_t)& =\left.\langle Y,T\rangle \right|{}_a^b-\int_a^b \left \langle Y,\frac{dT}{ds}\right\rangle ds \\ \left.\frac{d}{dt}\right|{}_{t=0}\mathcal{B}(\gamma_t) &= \left.\left(\left\langle\frac{dY}{ds},\frac{dT}{ds}\right\rangle-\left\langle Y,\frac{d^2T}{ds^2}+\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle T\right\rangle \right)\right|{}_a^b \\ &\quad +\int_a^b \left(\left\langle Y,\frac{d^3T}{ds^3} +3 \left\langle\frac{dT}{ds},\frac{d^2T}{ds^2}\right\rangle T +\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle \frac{dT}{ds}\right\rangle\right)ds.\end{aligned} $$

Proof

By Theorem 2.8,

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0}\mathcal{B}(\gamma_t) &= \int_a^b\left(\frac{1}{2}\left\langle \frac{dT}{ds}, \frac{dT}{ds}\right\rangle ds\right)^{\mbox{\bf \hspace{-0.35ex}.}} \\ &= \int_a^b\left(\left\langle\frac{d^2Y}{ds^2},\frac{dT}{ds}\right\rangle -\frac{3}{2}\left\langle \frac{dY}{ds},T\right\rangle\left\langle \frac{dT}{ds},\frac{dT}{ds}\right\rangle\right)ds \\ &= \int_a^b\left( \frac{d}{ds}\left\langle\frac{dY}{ds},\frac{dT}{ds}\right\rangle - \left\langle \frac{dY}{ds},\frac{d^2T}{ds^2}\right\rangle \right. -\frac{3}{2}\frac{d}{ds}\left(\langle Y,T\rangle \left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle\right) \\ &\qquad +\left.\frac{3}{2}\left\langle Y,\frac{dT}{ds}\right\rangle \left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle+3 \langle Y,T\rangle \left\langle\frac{dT}{ds},\frac{d^2T}{ds^2}\right\rangle\right) ds \\ &=\int_a^b \frac{d}{ds}\left(\left\langle\frac{dY}{ds},\frac{dT}{ds}\right\rangle-\left\langle Y,\frac{d^2T}{ds^2}\right\rangle -\frac{3}{2}\langle Y,T\rangle \left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle\right)ds \\ &\qquad +\int_a^b \left(\left\langle Y,\frac{d^3T}{ds^3} +3 \left\langle\frac{dT}{ds},\frac{d^2T}{ds^2}\right\rangle T +\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle \frac{dT}{ds}\right\rangle\right)ds\\ &=\left.\left(\left\langle\frac{dY}{ds},\frac{dT}{ds}\right\rangle-\left\langle Y,\frac{d^2T}{ds^2}+\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle T\right\rangle \right)\right|{}_a^b \\ &\qquad +\int_a^b \left(\left\langle Y,\frac{d^3T}{ds^3} +3 \left\langle\frac{dT}{ds},\frac{d^2T}{ds^2}\right\rangle T +\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle \frac{dT}{ds}\right\rangle\right)ds. \end{aligned} $$

□

3 Critical Points of Length and Bending Energy

Variations of curves (as defined in Definition 2.3) are needed in order to define and determine those curves that represent equilibria of geometrically interesting variational functionals. Functionals are certain real-valued functions on the space $\mathcal {M}$ of all curves $\gamma \colon [a,b]\to \mathbb {R}^n$, that was already introduced in Sect. 2.1.

Definition 2.10

Suppose we have a way to assign to each curve $\gamma \colon [a,b]\to \mathbb {R}^n$ a real number $\mathcal {E}(\gamma )$. Then $\mathcal {E}$ is called a smoothfunctional if for every smooth one-parameter family

$$\displaystyle \begin{aligned} t\mapsto \gamma_t,\quad t\in [t_0,t_1]\end{aligned}$$

of curves $\gamma _t\colon [a,b]\to \mathbb {R}^n$ the function

$$\displaystyle \begin{aligned} [t_0,t_1]\to \mathbb{R},\ t\mapsto \mathcal{E}(\gamma_t)\end{aligned} $$

is smooth.

In many circumstances, we want to consider only variations of $\gamma \colon [a,b]\to \mathbb {R}^n$ that keep the curve fixed near the boundary of the interval $[a,b]$ (see Fig. 2.2).

Fig. 2.2

A variation $\gamma _t$ of a curve $\gamma$ with support in the interior

Definition 2.11

Let $\gamma \colon [a,b]\to \mathbb {R}^n$ a curve. Then a variation

$$\displaystyle \begin{aligned} t\mapsto \gamma_t,\quad t\in [t_0,t_1]\end{aligned}$$

of $\gamma$ is said to have support in the interior of $[a,b]$ if there is $\epsilon >0$ such that for all $x\in [a,a+\epsilon ]\cup [b-\epsilon ,b]$ we have

$$\displaystyle \begin{aligned} \gamma_t(x)=\gamma(x)\quad \text{for all}\quad t\in [t_0,t_1].\end{aligned}$$

Now we can make precise what we meant by an equilibrium of a variational energy:

Definition 2.12

Let $\mathcal {E}$ be a smooth functional on the space of curves $\gamma \colon [a,b]\to \mathbb {R}^n$. Then a curve $\gamma \colon [a,b]\to \mathbb {R}^n$ is called a critical point of $\mathcal {E}$ if for all variations $t\mapsto \gamma _t$ of $\gamma$ with support in the interior of $[a,b]$ we have

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \mathcal{E}(\gamma_t)=0.\end{aligned}$$

We denote the space

$$\displaystyle \begin{aligned} \left\{ Y\colon\!\![a,b]\rightarrow\mathbb{R}^n\, \text{smooth}\, {\big\vert} \left.Y\right|{}_{[a,a+\delta]\cup[b-\delta,b]}= 0 \ \, \text{for some}\ \, \delta>0\right\} \end{aligned} $$

of all functions $Y\colon [a,b]\to \mathbb {R}^n$ with support in the interior of $[a,b]$ (Definition A.4) by $C^\infty _0((a,b),\mathbb {R}^n)$.

Theorem 2.13

For every vector field$Y\colon [a,b]\to \mathbb {R}^n$along a curve$\gamma \colon [a,b]\to \mathbb {R}^n$there is a variation$t\mapsto \gamma _t$with variational vector field Y . If Y  has support in the interior of$[a,b]$, then also the variation$t\mapsto \gamma _t$can be chosen in such a way that it has support in the interior of$[a,b]$.

Proof

The proof of Theorem 2.13 is left as an exercise. □

Remark 2.14

In the case of the length functional, instead of using variations with support in the interior we could have used variations that fix both end points. For other variational problems (that involve higher derivative), additional derivatives (not only the function value) of $\gamma$ would have to be clamped to fixed values at the end points. On the other hand, variations with support in the interior will work all the time, with equivalent results.

Theorem 2.15 (Fundamental Lemma of the Calculus of Variations)

On the vector space$C^\infty \left ([a,b],\mathbb {R}^n\right )$equipped with the inner product

$$\displaystyle \begin{aligned} \langle\!\langle f,g\rangle\!\rangle := \int_a^b\langle f,g \rangle\end{aligned}$$

only the zero vector is in the orthogonal complement of$C^\infty _0\left ((a,b),\mathbb {R}^n\right )$:

$$\displaystyle \begin{aligned} C^\infty_0\left((a,b),\mathbb{R}^n\right)^\perp=\lbrace 0 \rbrace.\end{aligned}$$

Proof

Suppose that $f\in C^\infty \left ([a,b],\mathbb {R}^n\right )$ would be non-zero but in the orthogonal complement $C^\infty _0\left ((a,b),\mathbb {R}^n\right )^\perp$. Then there would be $x_0\in [a,b]$ such that $f(x_0)\neq 0$. Choose $\delta >0$ such that $[x_0-\delta ,x_0+\delta ]\subset (a,b)$ and $\langle f(x),f(x_0)\rangle >0$ for all $x\in [x_0-\delta ,x_0+\delta ]$. Construct a smooth bump function (cf. Appendix A.2)

$$\displaystyle \begin{aligned} g\in C^\infty_0((x_0-\delta,x_0+\delta),\mathbb{R}^n)\subset C_0^\infty((a,b),\mathbb{R}^n)\end{aligned}$$

such that $g\geq 0$ and $g(x_0)=1$. Then $\langle f,g\rangle \neq 0$, which implies $f\notin C^\infty _0\left ((a,b),\mathbb {R}^n\right )^\perp$, a contradiction. □

Now we are in the position to determine the critical points of the length functional:

Theorem 2.16

A curve$\gamma \colon [a,b]\to \mathbb {R}^n$is a critical point of the length functional$\mathcal {L}$if and only if its unit tangent field$T\colon [a,b]\to \mathbb {R}^n$is constant, i.e. if$\gamma$parametrizes a straight line segment.

Proof

By Theorem 2.9 and 2.15, $\gamma$ is a critical point of $\mathcal {L}$ if and only if for all $Y\in C^\infty _0\left ((a,b),\mathbb {R}^n\right )$ we have

$$\displaystyle \begin{aligned} \left\langle\left\langle Y,\frac{dT}{ds}\right\rangle\right\rangle =0.\end{aligned}$$

By Theorem 2.15 this is the case if and only if $\frac {dT}{ds}=0$. □

Definition 2.17

A curve $\gamma \colon [a,b]\to \mathbb {R}^n$ is called a free elastic curve if it is a critical point of the bending energy functional $\mathcal {B}$.

An almost identical proof as the one of Theorem 2.16 gives us

Theorem 2.18

A curve$\gamma \colon [a,b]\to \mathbb {R}^n$is a free elastic curve if and only if its unit tangent field$T\colon [a,b]\to \mathbb {R}^n$satisfies

$$\displaystyle \begin{aligned} \frac{d^3T}{ds^3} +3 \left\langle\frac{dT}{ds},\frac{d^2T}{ds^2}\right\rangle T +\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle \frac{dT}{ds}=0\end{aligned}$$

or equivalently

$$\displaystyle \begin{aligned} \frac{d^4\gamma}{ds^3} +3 \left\langle\frac{d^2\gamma}{ds^2},\frac{d^3\gamma}{ds^3}\right\rangle \frac{d\gamma}{ds} +\frac{3}{2}\left\langle\frac{d^2\gamma}{ds^2},\frac{d^2\gamma}{ds^2}\right\rangle \frac{d^2\gamma}{ds^2}=0.\end{aligned}$$

Solving the fourth order differential equation for $\gamma$ that appears in Theorem 2.18 with suitable initial values will give us unit speed parametrizations of free elastic curves. In the next chapter we will explore in more detail the geometric consequences of this differential equation.

4 Constrained Variation

In the context of many variational problems that arise in applications, general variations might violate some constraints that are imposed by the nature of the problem at hand. For example, thin elastic wires (for most practical purposes) have a fixed length. This means that here we should minimize bending energy only among those curves (held fixed near their boundary) that have a prescribed length.

This kind of problem is known under the name of optimization under constraints. Here we will work with a definition of a critical point under constraints that is slightly stronger than the standard one. The usual definition would replace the condition $\left .\frac {d}{dt}\right |{ }_{t=0} \tilde {\mathcal {E}}=0$ by the requirement that $\tilde {\mathcal {E}}(\gamma _t)$ is independent of t.

Definition 2.19

Let $\mathcal {E},\tilde {\mathcal {E}}$ be two smooth functionals on the space of all curves $\tilde {\gamma }\colon [a,b]\to \mathbb {R}^n$. Then a curve $\gamma \colon [a,b]\to \mathbb {R}^n$ is called a critical point of$\mathcal {E}$under theconstraint of fixed $\tilde {\mathcal {E}}$, if for all variations $t\mapsto \gamma _t$ of $\gamma$ with support in the interior of $[a,b]$

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \tilde{\mathcal{E}}=0\end{aligned}$$

implies

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \mathcal{E}=0.\end{aligned}$$

Both for the length functional $\mathcal {E}=\mathcal {L}$ and for the bending energy $\mathcal {E}=\mathcal {B}$ we know (Theorems 2.8 and 2.9) how to express the infinitesimal variation of $\mathcal {E}$ that corresponds to a variational vector field $Y\colon [a,b]\to \mathbb {R}^n$ with support in the interior of $[a,b]$ as an integral

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \mathcal{E}=\int_a^b \langle Y,G_\gamma\rangle\end{aligned}$$

for some smooth map $G_\gamma \colon [a,b]\to \mathbb {R}^n$. If a formula like the one above holds, $G_{\gamma }$ is called the gradient of the energy $\mathcal {E}$ at $\gamma$.

Theorem 2.20

Let$\mathcal {E},\tilde {\mathcal {E}}$be two smooth functionals on the space of all curves$\tilde {\gamma }\colon [a,b]\to \mathbb {R}^n$. Suppose we have a way to associate to each curve$\gamma \colon [a,b]\to \mathbb {R}^n$smooth maps

$$\displaystyle \begin{aligned} G_\gamma,\widetilde{G}_\gamma\colon[a,b]\to \mathbb{R}^n\end{aligned}$$

such that for all variations$t\mapsto \gamma _t$of$\gamma$with support in the interior of$[a,b]$we have

Then$\gamma$is a critical point of$\mathcal {E}$under the constraint of fixed$\tilde {\mathcal {E}}$if and only if there is a constant$\lambda \in \mathbb {R}$such that

$$\displaystyle \begin{aligned} G_\gamma=\lambda \widetilde{G}_\gamma.\end{aligned}$$

$\lambda$is called aLagrange multiplierfor the constraint of fixed$\tilde {\mathcal {E}}$.

Proof

We apply Theorem 2.21 below to the case where $H=C^\infty \left ([a,b],\mathbb {R}^n\right )$, $V=C_0^\infty \left ((a,b),\mathbb {R}^n\right )$ and $U=\mathbb {R}\tilde {G}_\gamma$. Then $\gamma$ is a critical point of $\mathcal {E}$ under the constraint of fixed $\tilde {\mathcal {E}}$ if and only if $G_\gamma$ is orthogonal to all Y  that are simultaneously in V  and orthogonal to U, i.e

$$\displaystyle \begin{aligned} G_\gamma \in (V\cap U^\perp)^\perp =U=\mathbb{R}\widetilde{G}_\gamma.\end{aligned}$$

□

The theorem below is pure linear algebra, no Functional Analysis is involved. The formulation is such that it can also be applied to a situation where there are constraint functionals $\mathcal {E}_1,\ldots ,\mathcal {E}_k$ instead of a single functional $\tilde {\mathcal {E}}$.

Theorem 2.21

Let H be a (possibly infinite dimensional) vector space with inner product$\langle .,.\rangle$. Let$V\subset H$be a subspace such that$V^\perp =\lbrace 0 \rbrace$and$U\subset H$finite dimensional. Then

$$\displaystyle \begin{aligned} (U^\perp\cap V)^\perp = U.\end{aligned}$$

Proof

As for all $x\in U^\perp \cap V$ it holds that $\langle u,x\rangle =0$ for all $u\in U$, the inclusion $U\subset (U^\perp \cap V)^\perp$ is immediate. In order to show that also $(U^\perp \cap V)^\perp \subset U$ we choose an orthonormal basis $\lbrace u_1,\ldots ,u_n\rbrace$ of U and define the map

$$\displaystyle \begin{aligned} P\colon H\rightarrow U,\ x\mapsto \sum_{i=1}^n\langle x,u_i\rangle u_i.\end{aligned}$$

It is not hard to check that P defines an orthogonal projection of H onto U, i.e. $P^2=P$, $P^\ast =P$ and $\mathrm {im}\,P = U$. Now for $u\in U$ and $h\in H$ it holds

$$\displaystyle \begin{aligned} \langle u,h\rangle=\langle P(u),h\rangle=\langle u, P(h)\rangle.\end{aligned}$$

Therefore we have $U\cap P(V)^\perp \subset V^\perp =\lbrace 0\rbrace$, hence $P(V)=U$. So there are $v_1,\ldots ,v_n\in V$ such that $P(v_i)=u_i$. We now define the map

$$\displaystyle \begin{aligned} Q\colon H\rightarrow V,\ x\mapsto\sum_{i=1}^n\langle x,v_i\rangle v_i\end{aligned}$$

which is symmetric (i.e. $Q^\ast =Q$) and satisfies $\mathrm {im}\,Q\subset V$ and $P\circ Q\big \vert _U=\mathrm {id}_U$. Therefore, for $x\in (U^\perp \cap V)^\perp$ and $v\in V$:

$$\displaystyle \begin{aligned} \langle x-P\circ Q(x),v\rangle=\langle x,v-Q\circ P(v)\rangle=0,\end{aligned}$$

since $v-Q\circ P(v)\in U^\perp \cap V$. Thus $x-P\circ Q(x)\in V^\perp =\lbrace 0 \rbrace$ and therefore $x=P\circ Q(x)\in U$. □

Definition 2.22

A curve $\gamma \colon [a,b]\to \mathbb {R}^n$ is called a torsion-free elastic curve if it is a critical point of bending energy under the constraint of fixed length (Fig. 2.3).

Fig. 2.3

Elastic curves are everywhere

Theorems 2.8, 2.9 and 2.20 together allow us to characterize torsion-free elastic curves by a differential equation:

Theorem 2.23

A curve$\gamma \colon [a,b]\to \mathbb {R}^n$is a torsion-free elastic curve if and only if there is a constant$\lambda \in \mathbb {R}$such that its unit tangent field satisfies

$$\displaystyle \begin{aligned} \frac{d^3T}{ds^3} +3 \left\langle\frac{dT}{ds},\frac{d^2T}{ds^2}\right\rangle T +\frac{3}{2}\left\langle\frac{dT}{ds},\frac{dT}{ds}\right\rangle \frac{dT}{ds}-\lambda \frac{dT}{ds}=0\end{aligned}$$

or, equivalently,

$$\displaystyle \begin{aligned} \frac{d^4\gamma}{ds^4} +3 \left\langle\frac{d^2\gamma}{ds^2},\frac{d^3\gamma}{ds^3}\right\rangle \frac{d\gamma}{ds} +\frac{3}{2}\left\langle\frac{d^2\gamma}{ds^2},\frac{d^2\gamma}{ds^2}\right\rangle \frac{d^2\gamma}{ds^2}-\lambda \frac{d^2\gamma}{ds^2} =0.\end{aligned}$$

The constant$\lambda$is called thetensionof$\gamma$.

5 Torsion-Free Elastic Curves and the Pendulum Equation

By Theorem 1.16 every curve in $\mathbb {R}^n$ admits a reparametrization $\gamma \colon [0,L]\to \mathbb {R}^n$ with unit speed. Then for any function $g\colon [0,L]\to \mathbb {R}^k$ the derivative with respect to arclength is just the ordinary derivative:

Theorem 2.24

A curve$\gamma \colon [0,L]\to \mathbb {R}^n$with unit speed is torsion-free elastic with tension$\lambda$if and only if its unit tangent field$T\colon [a,b]\to S^{n-1}$solves the equation of motion

$$\displaystyle \begin{aligned} T^{\prime\prime}-\langle T^{\prime\prime},T\rangle T = \mathbf{a}-\langle \mathbf{a},T\rangle T\end{aligned}$$

of a spherical pendulumwith unit mass and somegravity vector$\mathbf {a}\in \mathbb {R}^n$and$\lambda$equals the total energy of the pendulum:

$$\displaystyle \begin{aligned} \lambda = \frac{1}{2}\langle T',T'\rangle -\langle \mathbf{a},T\rangle.\end{aligned}$$

Proof

Let $\gamma \colon [0,L]\to \mathbb {R}^n$ be a torsion-free elastic curve with tension $\lambda$ and with unit speed. Then, by Theorem 2.23

$$\displaystyle \begin{aligned} 0 = T^{\prime\prime\prime}+3\langle T', T^{\prime\prime}\rangle T+\frac{3}{2}\langle T',T'\rangle T'-\lambda T^\prime=\left(T^{\prime\prime}+\frac{3}{2}\langle T',T'\rangle T -\lambda T \right)',\end{aligned}$$

i.e. if there is a constant vector $\mathbf {a}\in \mathbb {R}^n$ such that

$$\displaystyle \begin{aligned} T^{\prime\prime}+\frac{3}{2}\langle T',T'\rangle T-\lambda T=\mathbf{a}.\end{aligned}$$

Looking at the component orthogonal to T on both sides of this equation gives us the first of the two equations that we want to prove. Taking the scalar product with T and using

$$\displaystyle \begin{aligned} 0 =\frac{1}{2}\langle T,T\rangle^{\prime\prime}=\langle T^{\prime\prime},T\rangle +\langle T',T'\rangle\end{aligned}$$

we obtain the second equation. Conversely, if $T\colon [0,L]\to S^{n-1}$ solves the pendulum equation

$$\displaystyle \begin{aligned} T^{\prime\prime}-\langle T^{\prime\prime},T\rangle T = \mathbf{a}-\langle \mathbf{a},T\rangle T,\end{aligned}$$

then it is easy to verify that the total energy $\lambda$ defined by

$$\displaystyle \begin{aligned} \lambda = \frac{1}{2}\langle T',T'\rangle -\langle \mathbf{a},T\rangle\end{aligned}$$

is constant and

$$\displaystyle \begin{aligned} T^{\prime\prime}+\frac{3}{2}\langle T',T'\rangle T-\lambda T=\mathbf{a}.\end{aligned}$$

□

Figure 2.4 shows planar torsion-free elastic curves that lie in a plane. They arise from pendulum motion on a circle, whereas a pendulum motion on $S^2$ gives a torsion-free elastic curve in $\mathbb {R}^3$ as seen in Fig. 2.5.

Fig. 2.4

Trajectories of a pendulum (drawn in blue color). Below each of these trajectories the corresponding torsion-free elastic curve is shown

Fig. 2.5

Trajectory of a pendulum on $S^2$ (drawn in blue color), together with the corresponding torsion-free elastic curve in $\mathbb {R}^3$