Question
I am interested in solving for $u : S^1 \to S^1$ given the following ODE, where $v : S^1 \to S^1$ is a given continuous function:
$$u'' = -A\cos(u-v) \tag{$\dagger$}$$
Is there any hope of an analytic solution? Can anything be said about the behavior of the solution as $A$ varies? If it helps, I have $\int_{S^1} \cos[v(\theta)]\ d\theta = \int_{S^1} \sin[v(\theta)]\ d\theta = 0$.
Background
I've been interested in crystalline and anisotropic mean curvature flows, along the line of the work of Novaga and Chambolle, e.g. https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.21668
My actual problem is more involved and three-dimensional, but to build intuition I've been trying to think about the following two-dimensional simplification.
Consider a curve $\gamma:S^1 \to \mathbb{R}^2$, which for simplicity we can take to be arclength-parameterized with arc length $2\pi$. Let us write $\gamma'(\theta) = [\cos(v[\theta]), \sin(v[\theta])]$. I want to study sections of the bundle of directors $\hat{m}(\theta) \in S^1$ along the curve: I want these directors to approximate the orientation of the curve normal, while also behaving more smoothly near regions of sharp curvature. Let $\hat{n}(\theta) = [-\sin(v[\theta]), \cos(v[\theta])]$ denote the curve normal at $\theta$; I would like to look at minimizers of the functional $$E(\hat m) = \int_{S^1} \left(\frac{1}{2}\left\|\hat{m}'(\theta)\right\|^2 + \frac{A}{2} \left\|\hat m(\theta) - \hat n(\theta)\right\|^2\right)\,ds$$ where the real constant $A$ controls the relative importance of smoothness vs adapting to the curve normal.
Applying the calculus of variations, we have that an extremizer $\hat{m}^*(\theta) = [\cos u^*(\theta), \sin u^*(\theta)]$ of $E$ satisfies the ODE $(u^*)''(\theta) = -A\cos[u^*(\theta) - v(\theta)]$, where $v: S^1\to S^1$ is prescribed by the choice of $\gamma$. And we know that $\int_{S^1} \cos[v(\theta)]\ d\theta = \int_{S^1} \sin[v(\theta)]\ d\theta = 0$, since the curve $\gamma$ is closed. Hence my question.
My eventual goal is to take variations of $E(\hat{m}^*)$ with respect to $\gamma$ to flow the curve, so an analytic solution to the ODE ($\dagger$) would be very helpful.
Some trivial observations: $E(\hat{m}^*)$ is bounded above by on the one hand $A\pi$, and on the other hand, $\int_{S^1} \kappa(\theta)^2\,ds$ (where $\kappa(\theta)$ is the curvature at $\theta$), for a constant director field, and $\hat m = \hat n$, respectively.
For arbitrary $v$? No, no hope. For certain specific $v$, yes. In fact, let $U(\theta)$ be any twice-differentiable function with $|U''/A|< 1$ and take $v(\theta) = U(\theta) + \arccos(U''/A)$, and $U(\theta)$ will be a solution.