Full time derivative of the Frank-Oseen energy. Mathematical problem

Question

I am studying liquid crystal theory with the book Kleman, Lavrentovich, Soft Matter Physics.

In the Ericksen-Leslie theory, Frank-Oseen energy density is:

$$ f=0.5*(K_1*div^2 (n)+K_2 *(n*curl(n))^2+K_3*(n \times curl(n) )^2) $$

Later they take a full derivative with respect to time:

$$ {{df} \over {dt}}= {\partial f \over \partial n_i} {dn_i \over dt} + {\partial f \over \partial (\partial n_i / \partial x_j)} {d \over dt} {\partial n_i \over \partial x_j} $$

Here $\overrightarrow{n}$ is director, and summation over repeated indexes is used. $K_1,K_2,K_3$ are constants. $x,y,z = x_1, x_2,x_3$ respectively are spatial variables.

So, as you see, $\overrightarrow{n}=\overrightarrow{n}(x,y,z,t)$ and $f$ is the function of $n$

The above formula for $df/dt$ looks like the famous formula for the total derivative, but if the independent variables are $t$ and spatial derivatives of the director components (9 terms instead of 3).

How can I make sure (proove), that I indeed can use above variables as independent, instead of standard ones $t,x,y,z$.

P.S. As far as I understand, there are 10 terms in the formula for $df/dt$ , although with usual set of independent variables $t,x,y,z$ there are only 4 terms

Edit: In order to make things clear: director is a vector with reversal symmetry. For this problem you can treat $\overrightarrow{n}$ as usual vector field, like $\overrightarrow {E}$ in electrostatics.

Answer 1

Comments to the question (v7):

1. The director $\vec{n}(\vec{r},t)$ is a vector-valued field. Ericksen-Leslie theory is a field theory.

2. Before studying variational calculus in field theory, and asking which variables are independent, and which are not, it is highly recommended to understand the corresponding problem in point mechanics, see e.g. this Phys.SE post.

3. Now the main lessons from point mechanics applies to field theory as well. E.g. assuming no explicity spacetime dependence, the Frank-Oseen energy density ${\cal F}(\vec{n}, T)$ is a function $$\tag{1} {\cal F}:\mathbb{R}^3 \times \mathbb{R}^9~\cong~ \mathbb{R}^{12} \longrightarrow \mathbb{R} $$ of a vector $\vec{n}\in\mathbb{R}^3 $ and a tensor $T\in\mathbb{R}^9$. These 12 arguments are independent variables in the Frank-Oseen energy density ${\cal F}$.

4. The independence of $\vec{n}$ and $T$ in ${\cal F}(\vec{n}, T)$ is similar to the independence of $q$ and $v$ in the Lagrangian $L(q,v,t)$ for point mechanics.

5. It may seem a bit pedantic to introduce the tensor field $T_{ij}$, but this is e.g. to make the notion of partial derivatives $$ \tag{2}\frac{\partial {\cal F}}{\partial T_{ij}} $$ well-defined and clear.

6. Next we substitute $$\tag{3} T~=~\vec{\nabla}\vec{n}$$ in ${\cal F}$, as one would expect, and consider ${\cal F}(\vec{n},\vec{\nabla}\vec{n})$.

7. Let $\vec{v}=\frac{d\vec{n}}{dt}$ be the corresponding velocity field to $\vec{n}$.

8. OP correctly identifies the formula for the total time derivative.

9. Now imagine that we know the initial spatial distribution of $$\tag{4}\vec{r}~\mapsto~ \vec{n}(\vec{r},t_0)\quad\text{and}\quad\vec{r}~\mapsto~ \vec{v}(\vec{r},t_0)$$ for some initial time $t_0$, i.e. we are given appropriate Cauchy data$^1$. Imagine furthermore, that we would like to find the distribution $\vec{n}(\cdot,t)$ for some future $t$. Note in particular that the corresponding initial tensor field configuration $$\tag{5}\vec{r}~\mapsto~T(\vec{r},t_0)~=~\vec{\nabla}\vec{n}(\vec{r},t_0)$$ is not independent of (and in fact completely determined by) the initial conditions (4).

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$^1$ Here we assume that the exists an appropriate 2nd order time evolution equation for this problem.