Lecture 08 - Elastica 2

#math

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

If $w$ is a symmetry we have:

\begin{matrix} (*) & γ^{'} \cdot w^{'} - J \cdot w = const \end{matrix}

We have previously implemented translational symmetry. We are now interested in studying rotational symmetry:

Let $W = γ \times W_{0}$ , the differentiating and using $(*)$ it gives us:

κ N \cdot T \times W_{0} - J \cdot γ \times W_{0} = const

Note by property of dot product and cross product ( $A \cdot B \times C = C \cdot A \times B = A \times B \cdot C$ ) we obtain:

\begin{aligned} (κ N \times T - J \times γ) \cdot W_{0} & = const \\ κ B + J \times γ & = A, where A is a constant vector \\ ⟹ κ B & = A + γ \times J \end{aligned}

So the vector field:

I = κ B = A + γ \times J

is a restriction of an isometry to the curve.
The vector fields $J$ and $I$ play an important role in the elastica.

Definition

A killing vector field along a curve is a vector field along the curve which is the restriction of an infinitesimal isometry of the ambient space.

If $γ$ is an elastica in $R^{3}$ then we have two killing fields $J$ and $I$ .

Let $γ (s)$ be the center line of the rod we are studying, then we have the material frame:

Γ = {γ (s) : T (s), M_{1} (s), M_{2} (s)}

If we have a Darboux vector $Ω = m T - m_{2} M_{1} + m_{1} M_{2}$ then:

{(\begin{matrix} T \\ M_{1} \\ M_{2} \end{matrix})}^{'} = (\begin{matrix} 0 & m_{1} & m_{2} \\ - m_{1} & 0 & m \\ - m_{2} & - m & 0 \end{matrix})

Total elastic energy is given by $E (γ) = \frac{1}{2} \int α_{1} (m_{1})^{2} + α_{2} (m_{2})^{2} + β m^{2} d s$ .
Symmetric case of $α_{1} = α_{2}$ we get:

E (γ) = \frac{1}{2} \int α (m_{1}^{2} + m_{2}^{2}) + β m^{2} d s

The first term with the $α$ 's refers to the bending energy and the term with the $β$ refers to the twisting energy.

Now wrt to the Bishop frame of the curve we get:

\begin{aligned} M_{1} & = U c o s (θ) + V s i n (θ) \\ M_{2} & = - U s i n (θ) + V c o s (θ) \\ m & = θ^{'} \end{aligned}

where $θ$ is the angle between $M_{1}$ and $U$ . Thus:

E = \frac{1}{2} \int α κ^{2} + β {θ^{'}}^{2} d s

For a curve/elastic rod:

F (γ) = λ_{1} \int_{γ} d s + λ_{2} \int_{γ} τ d s + λ_{3} \int_{γ} κ^{2} d s

As usual the extremum of this will give us our our equations of motion.

We first define the bending energies:

E (γ) = \int_{0}^{L} κ^{2} d s = \int_{0}^{L} < T^{'}, T^{'} > d s = E (T)

Since the length of the tangent is unit then one can consider it as a function that assigns to a point on the curve to a point on the unit circle that is: $$T: [0,L] \rightarrow S^{2}$$
Note we have a constraint that $\int_{0}^{L} T d s$ is constant. We can also rewrite it to be:

\sum a_{i} \int_{0}^{L} T_{i} d s = \int_{0}^{L} < a, T > d s

Thus we want to vary:

E^{λ} (T) = \int_{0}^{L} [< T^{'}, T^{'} > + 2 < a, T > + Λ (s) (< T, T > - 1)]

but instead we can simply just use the Euler Lagrange equation ( $\frac{d}{d s} \nabla_{T^{'}} L - \nabla_{T} L = 0$ ) which gives us:

T^{″} = Λ (s) T + a

Thus we obtain three equivalent representations:

\begin{aligned} T^{″} \times T & = a \times T \\ γ^{″} \times γ^{'} + Λ γ^{'} & = a \times γ + b \end{aligned}

If $ω$ is the angular velocity of the frame then:

\begin{aligned} T^{'} & = ω \times T \\ ω^{'} & = (1 - α) τ T^{'} + T \times a \end{aligned}

Applications to Biopolymers

Polymers are big molecules and because of their size, the physical behaviour of individual polymer molecules is drastically different than that of their small-molecule analogues. We want to study the dynamics of a wormlike chain. A semiflexible biopolymer is represented as a slender elastic rod with a elastic modulus characterizing the bending energy of the polymer. The axis of the polymer is denoted by space curve.

We consider a wormlike chain of length $L$ . The chain trajectory defines a space curve $γ (s)$ , and the chain contour is assumed inextensible which ensures $| T (s) | = | \partial_{s} γ (s) | = 1$ .

The energetics of the wormlike chain model is given by a bending Hamiltonian of the form:

β H = \frac{l_{p}}{2} \int_{0}^{L} d s (γ_{s s})^{2},

where $β = 1 / k_{B} T$ and $l_{p}$ is the persistence length of the free polymer chain.

We can rewrite the Hamiltonian in the following form:

H = \frac{C}{2} \int t_{s}^{2} d s = \frac{C}{2} \int κ^{2} d s,

where $C$ is constant. This precisely the elastic energy of the rod in the wormlike chain model, where $C$ stands for the bending modulus.

Let us scale out of the constant $C$ and use variational principle $δ H = 0$ on the Hamiltonian:

H = \frac{1}{2} \int [t_{s}^{2} - λ (t^{2} - 1)] d s

Then the corresponding time evolution for $t$ is given by:

t_{u} = t \times t_{s s}