Lecture 07 - Elastica 1

#math

We are going to be studying 'elastic' curves.

Definition

Bending energy of a curve (which will be a conserved quantity) is defined to be:

E_{l} = \frac{1}{2} \int_{0}^{L} κ^{2} d s

If we also want to twist the curve then:

{\hat{E}}_{l} = \frac{1}{2} \int_{0}^{L} κ^{2} d s + \int_{0}^{L} τ^{2} d s

We want to obtain the equations of motion from the bending energy of a curve.

Consider a regular curve in Euclidean space $R^{3}$ , defined on a fixed interval $[a_{1}, a_{2}]$ . (One can do the same on a Riemannian manifold instead by replacing derivatives with covariant derivatives instead).

Let $Ω = {γ : γ (a_{i}) = α_{i}, γ^{'} (a_{i}) = α_{i}^{'}}$ , now consider the subspace of $Ω$ such that:

Ω_{u} = {γ \in Ω : | | γ^{'} | | = 1}

Elastica minimizes the bending energy that is:

F = \frac{1}{2} \int_{γ} κ^{2} d s = \frac{1}{2} \int_{γ} | | γ^{″} | |^{2} d s

We are interested in varying $F$ subject to the constraint of the defined subspace, thus we define $F^{λ} (γ) = \frac{1}{2} \int | | γ^{″} | |^{2} + Λ (s) (| | γ^{'} | |^{2} - 1) d s$ , where $Λ$ is the langrange multiplier. Lagrange multiplier says that the minimum of $F$ on $Ω_{u}$ is a stationary point for $F^{λ}$ for some values of $Λ (s)$ .

Derivation of equations of motion

If $w$ be a vector along $γ$ :

\partial F^{λ} (w) = \frac{\partial}{\partial ϵ} F^{λ} (γ + ϵ w) |_{ϵ = 0}

We first compute the $F^{λ}$ on the right:

\begin{aligned} F^{λ} (γ + ϵ w) & = \frac{1}{2} \int_{a_{1}}^{a_{2}} | | (γ + ϵ w)^{″} | |^{2} + Λ (s) (| | (γ + ϵ w)^{'} | |^{2} - 1) d s \end{aligned}

Thus:

\partial_{ϵ} F^{λ} (γ + ϵ w) |_{ϵ = 0} = \int_{a_{1}}^{a_{2}} (γ^{″} \cdot w^{″} + Λ (s) γ^{'} \cdot w^{'}) d s = 0

Integrating by parts gives us:

\int_{a_{1}}^{a_{2}} [γ^{‴} - (Λ γ^{'})^{'}] \cdot w d s + (γ^{″} \cdot w^{″} - {(Λ γ^{'} - γ^{‴}) \cdot w) |}_{a_{1}}^{a_{2}}

Thus we obtain our Equations of motion:

\begin{aligned} E (γ) = γ^{‴} - (Λ γ^{'})^{'} & = 0 \\ (*) & ⟹ γ^{‴} - Λ γ^{'} & = J \end{aligned}

Da Rios Equation from the Elastic equations

\begin{aligned} γ^{'} & = T \\ ⟹ γ^{″} & = κ N \\ ∴ γ^{‴} & = - κ^{2} T + κ^{'} N + κ τ B \end{aligned}

Thus $(*)$ transforms to:

\begin{aligned} (- κ^{2} - Λ (s)) T + κ^{'} N + κ τ B & = J \end{aligned}

Differentiating gives us:

(- 3 κ κ^{'} - Λ^{'}) T - (κ^{″} - κ^{3} - Λ κ - κ τ^{2}) N + (κ τ^{'} + 2 κ^{'} τ) B = 0

Thus we obtain $Λ (s)$ by projecting the above equation on $T$ .

Λ (s) = \frac{- 3}{2} κ^{2} + \frac{λ}{2}

where $λ$ is some constant. Thus substituting back we obtain:

J = \frac{κ^{2} - λ}{2} T + κ^{'} N + κ τ B

Thus once again differentiating we obtain our Da Rios equation:

\begin{aligned} κ^{″} + \frac{1}{2} κ^{3} - κ τ^{2} - \frac{λ κ}{2} & = 0, \\ κ τ^{'} + 2 κ^{'} τ & = 0. \end{aligned}

We obtain from the second equation that $(κ^{2} τ)^{'} = 0$ implies $κ^{2} τ = c_{1}$ , thus reducing our system to:

κ^{″} + \frac{1}{2} κ^{3} - \frac{c_{1}^{2}}{κ^{3}} - λ \frac{κ}{2} = 0.

Notice that $2 J = κ^{2} - λ T + 2 κ^{'} N + 2 κ τ B$ , thus:

\begin{matrix} (**) & 4 | | J | |^{2} = (κ^{2} - λ)^{2} + 4 κ^{' 2} + 4 κ^{2} τ^{2} = a^{2} \end{matrix}

Taking $u = κ^{2}$ then $(* *)$ reduces to:

(u - λ)^{2} + \frac{u^{' 2}}{u} + 4 \frac{τ^{2}}{u} = a^{2}

Let $u^{' 2} = P (u)$ and multiply $u$ both sides of the equation then:

P (u) = - u^{3} + 2 λ u^{2} + (a^{2} - λ^{2}) u - 4 τ^{2}

Notice the following:

$P (0) = - 4 τ^{2} \leq 0$
$P (u) \to - \infty$ as $u \to \infty$
$P (u) \to \infty$ as $u \to - \infty$

Thus $P (u)$ has 3 real roots thus:

P (u) = - (u + α_{1}) (u + α_{2}) (u + α_{3})

Thus:

\begin{aligned} 4 λ & = α_{3} + α_{2} - α_{1} \\ a^{2} - λ^{2} & = α_{1} α_{3} + α_{1} α_{2} - α_{2} α_{3} \\ 4 τ^{2} & = α_{1} α_{2} α_{3} \end{aligned}

After some laborious calculation:

u (s) = α_{3} (1 - q^{2} s n^{2} (r s, p))

where:

\begin{aligned} p^{2} & = - \frac{α_{3} - α_{2}}{α_{3} + α_{2}} & q^{2} & = \frac{α_{3} - α_{2}}{α_{3}} \\ α_{3} & = k_{0}^{2} & r & = \frac{1}{2} \sqrt{α_{3} + α_{1}} \\ ω^{2} & = \frac{p^{2}}{q^{2}} & κ^{2} & = κ_{0}^{2} (1 - \frac{p^{2}}{ω^{2}} s n^{2} (\frac{k_{0}}{2 ω} s, p) \end{aligned}