You want to travel from one side of a quicksand $(1,0)$ pit to another side of the quicksand pit $(-1,0)$. The speed you can run is determined by how far you are away from the quicksand $|v(x,y)| = \sqrt{x^2+y^2}$. What is the optimal path you should travel such that the time taken to travel is minimized?
In my attempted solution:
- Use cylindrical coordinates $(r,\theta)$ and find an expression for the total time taken $T$
- Attempt to minimize $T$ using Beltrami's identity
- Solve the resulting differential equation to find $r(\theta)$
However, when I do this, I find $r = c_2 e^{c_1 \theta}$. This is strange because I would have expected a symmetrical path around the y axis. Also, when I plug in the start and end points I get nonsensical values for the constants $c_1$ and $c_2$. Can someone help me figure out what I'm doing wrong?
From
$$ T = \int_{0}^{\pi}\sqrt{1+\left(\frac{r'(\theta)}{r(\theta)}\right)^2}d\theta $$
we obtain the Euler-Lagrange equations
$$ r(\theta ) r''(\theta )-r'(\theta )^2=0 $$
with solution
$$ r(\theta) = c_2e^{c_1\theta} $$
now applying the boundary conditions
$$ \cases{ r(0) = c_2 = 1\\ r(\pi) = e^{\pi c_1}= 1 } $$
we determine $c_1 = 0, c_2 = 1$ so the minimum time orbit is the semi-circumference with radius $r = 1$