Given a "sufficiently well behaved", but not necessarily conservative, vector field ${\bf E}$ in $\mathbb{R}^3$ defined in a bounded domain $\Omega$, what is the maximum value of

$$ \int_C {\bf E} \cdot {\bf dl} $$

over all paths $C$ of lengths smaller than $L$.

I have tried a variational approach, with a Lagrange multiplier $\lambda$ for a fixed-length constraint, which results in the ODEs for the parametric path ${\bf x}(t)$:

$$ {\bf B} \times {\bf v}(t) = \lambda \frac{d {\bf v}}{dt}, \\ {\bf v} (t) = \frac{d{\bf x}}{dt} $$

where $ {\bf B} = \nabla \times \bf E$ is the curl of $\bf E$.

This is the equation of motion of a charged particle in a magnetic field. Since the force is orthogonal to the direction of motion, the magnitude of the velocity vector remains constant along the trajectory. So we can assume that $\|{\bf v}\| = 1$.

Stationarity conditions at the end points ${\bf x}_a$ and ${\bf x}_b$ yield:

$$ {\bf B \cdot v} + \lambda {\bf v} = 0, \quad \textrm{at} \, {\bf x}={\bf x}_a \textrm{ and } {\bf x}={\bf x}_b$$ which leads to

$$ \lambda = \|{\bf B}\|_{x=x_a}= \|{\bf B}\|_{x=x_b} \\ {\bf v} = {\bf B} / \|{\bf B}\| \quad \textrm{at} \, {\bf x}={\bf x}_a \textrm{ and } {\bf x}={\bf x}_b$$

This does not bring me significantly closer to finding a solution to the original problem, though, because the path obtained by integrating the ODEs (and hence the total work over of ${\bf E}$ along the path) depends extremely strongly on the choice of the starting point ${\bf x}_a$.

Does anyone have an idea of how to solve the original problem? Any clue in the existing literature?


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