I've come upon the following system of ODEs, where $f_1, f_2, f_3 : I \subset \mathbb{R}\to \mathbb{R}$:
$$f_1^2 + f_2^2 = 1$$
$$(f_1')^2 + (f_2')^2 + (f_3')^2 = 1$$
$$-f_2f_1'' + f_1f_2'' = (f_1'')^2 + (f_2'')^2 + (f_3'')^2 $$
I'm trying to find solutions (even trivial ones, really) but it hasn't been easy. What would be some useful restrictions on $f_1, f_2$ and $f_3$? If it's too much of a task to find particular solutions, how can one best understand this system?
Oh, and if anyone knows whether there's a way to tackle this using numerical methods I'd appreciate it too.
Best Answer
The first two equations say this is a motion on the cylinder with speed $1$. One family of trivial solutions is where the motion is in the $f_3$ direction only: $$ f_1(t) = \cos(\alpha),\ f_2(t) = \sin(\alpha), \ f_3(t) = t + c$$
EDIT: Following WalterJ's suggestion, if $f_1(t) = \cos(g(t))$ and $f2(t)=\sin(g(t))$, the first equation is satisfied automatically and the last two become $$ \eqalign{\dot{g}^2 + \dot{f_3}^2 &= 1 \cr \ddot{g} = \dot{g}^4 + \ddot{g}^2 + \ddot{f_3}^2\cr}$$
We can then eliminate $f_3$ from this: note that $$ \dot{f_3} \ddot{f_3} = - \dot{g} \ddot{g} $$ so $$\ddot{f_3}^2 = \dfrac{\dot{g}^2 \ddot{g}^2}{1-\dot{g}^2} $$
and thus we get one equation for $g$: $$ \ddot{g} = \dot{g}^4 + \ddot{g}^2 + \dfrac{\dot{g}^2 \ddot{g}^2}{1-\dot{g}^2} $$ Note this is a separable (but rather nasty) first-order equation in $\dot{g}$.