# Differential equation with some constraints

I'd like $\alpha,\beta,\gamma$ as functions of $t$, satisfying the following conditions: \begin{align} \alpha+\beta+\gamma & = 0 \\ \sin^2\alpha + \sin^2\beta + \sin^2\gamma & = c^2 \\ \left| \frac{d}{dt}(\sin\alpha,\sin\beta,\sin\gamma)\right| & = 1 \end{align} I'm thinking of $c^2$ as small. At the very least that means $<2$, and intuitively it means $\ll 2$. Some geometry shows that there is a qualitative change in the nature of the solutions when $c^2$ goes from $<2$ to $>2$.

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Would it be unsatisfactory to numerically compute the one-dimensional manifold defined by the first two constraints, and then compute a unit speed trajectory along that path? – Rahul Aug 27 '11 at 22:26
@Rahul: Not necessarily. But if that's all that can be done, it might be of interest to know that that's all that can be done. Also, if the solution is a function with interesting properties, maybe something could be said about those....... – Michael Hardy Aug 28 '11 at 1:34
Well, the simpler-looking problem $\lvert d(x,y)/dt \rvert = 1$ with $ax^2 + by^2 = 1$ is equivalent to finding the arc length parametrization of an ellipse, which is hard. So I'm not very hopeful for a nice solution to this problem, but maybe someone else has better ideas... – Rahul Aug 28 '11 at 2:46
I'm not sure your proposed "simpler-looking" problem is actually simpler. – Michael Hardy Aug 28 '11 at 15:06