I'm fairly new to the world of functional analysis, so I need a bit of help setting up my first variational problem. What I'm trying to do is to find a closed space curve of minimal torsion, that also respects a distance constraint.
I start with a parametric form of the curve, $r:\mathbb{R}\to \mathbb{R}^3$, whereby the torsion is given by $\tau = \frac{(r'\times r'')\cdot r'''}{||r'\times r''||^2}$. The Lagrangian for this problem is: $L = \int_{0}^{4\pi}\tau + \lambda(t)(D(r)-c)dt=\int_{0}^{4\pi}F(t,r,r',r'',r''')dt$, where $\lambda(t)$ are the Lagrange multipliers, $D(r)$ is a distance function, and $c$ is the desired clearance that the curve should maintain. The Euler-Lagrange equation for this problem is thus: $\frac{\partial F}{\partial r}-\frac{d}{dt}\frac{\partial F}{\partial r'}+\frac{d^2}{dt^2}\frac{\partial F}{\partial r''}-\frac{d^3}{dt^3}\frac{\partial F}{\partial r'''}=0$.
The question is how do I write the variational form of this equation, as numerical solvers require it. More precisely, how does the test function enter the equation, and what would be an advantageous form that could be used to compute it.
Thanks!