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 $>2$.
Differential equation with some constraints
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ordinary-differential-equations
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0Would 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? – 2011-08-27
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0@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....... – 2011-08-28
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1Well, 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... – 2011-08-28
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0I'm not sure your proposed "simpler-looking" problem is actually simpler. – 2011-08-28