I want to solve the problem: Find the curve satisfies following conditions.
- Minimize the functional $J$
- The coordinates of the start/end points are given
- Direction(tangential vector) at the start/end points are given
- The length of the curve is $L$
I parameterized the curve using $\theta(s)$. $s$ is the length from the start point, and $\theta$ is the angle of the tangential vector at $s$.
The functional $J$ is defined as following. $$ J[\theta] = \int_0^L f(\theta, \theta'; s)ds \qquad \left(\theta'\equiv \frac{d\theta}{ds}\right) $$
The curve $(x(s), y(s))$ can be writen using $\theta$ $$ x(s) = \int_0^s \cos(\theta(s))ds \\ y(s) = \int_0^s \sin(\theta(s))ds $$
Then, the second condition becomes following isoperimetric constraints. (set the start point as origin.) $$ x(L) = \int_0^L \cos(\theta)ds = p_x\\ y(L) = \int_0^L \sin(\theta)ds = p_y $$
condition 3, and 4 is already considered by the definitions.
Therefore the problem is equivalent to the euler lagrange equation with two isoperimetric constraints.
I wanted to verify this solution, so took it to my acquaintance. He said that the solution is wrong, but I can't understand.
Is there any wrong part?