I'm attempting to use a variational method to find a critical vector field. I want to find the local components of a given vector field, $\bf{v}$, that are as "close" to $\bf{v}$ as possible, but are also divergence-free. My thought is that I can search for a vector field of unit vectors, $\bf{u}$, such that the projection of $\bf{v}$ onto $\bf{u}$ is divergence free and the error between this projection and $\bf{v}$ is minimal. I can write this as a variational problem:
$$ \text{min}_{\textbf{u}} \int_V || \textbf{v} - (\textbf{v} \cdot \textbf{u})\textbf{u} || ^2 dV $$ $$ \nabla \cdot (\textbf{v} \cdot \textbf{u}) \textbf{u} = 0 $$ $$ \textbf{u} \cdot \textbf{u} = 1 $$
where $V$ is a smooth 2-D manifold. I can use Lagrange multipliers to remove the constraints and apply the divergence theorem to give the unconstrained variational problem:
$$ \text{min}_{\textbf{u}} \int_V || \textbf{v} - (\textbf{v} \cdot \textbf{u})\textbf{u} || ^2 - \nabla \lambda_1 \cdot \textbf{u} + \lambda_2 (\textbf{u} \cdot \textbf{u} - 1) dV $$ $$ \lambda_1 = 0 \quad \text{on } \partial V $$
At this point I'm not sure how to proceed. Deriving the Euler-Lagrange equations by setting $\textbf{u} = \textbf{u}^* + \epsilon \textbf{w}$ and setting the derivative with respect to $\epsilon$ to 0 at $\epsilon = 0$ gives a mess. I'm curious if anyone has seen a similar problem and what strategies can be used to solve this.
Thank you!