I want to minimize $I = \int |\dot{x}|^2 dt$ subject to the constraint $|x|^2=1$ (sphere) which gives an Euler equation of $\lambda x - \ddot{x} = 0$.
I have to show that the Euler equation is actually $|\dot{x}|^2 x - \ddot{x} = 0$. Is it right to assume that $\lambda=|\dot{x}|^2$ simply by the fact that it minimizes $I^* = \int |\dot{x}|^2- \lambda (|x|^2-1) dt$ which is $\geq 0$, so the $\lambda$ that minimizes $I^*$ is $|\dot{x}|^2$?
If I then try to integrate the Euler equation, then I get a SHM equation:
$x1= A \cos(|\dot{x}| t - C)$ where A, C are constants and similarly for x2, x3
But how do I combine them to give the equation of a great circle, since I don't know the $C$'s?
Thank you for any enlightenment!