[ODE] Theory Question: Where Does The Closed-Form Solution End?

[Gary R. Van Sickle]

Hi guys,

Under what conditions does the rigid body problem cease to have a
closed-form solution?  I know that there are a few almost-trivial cases
which have simple closed-form solutions, for instance
constant-linear-velocity, constant-linear-acceleration, the "rocket
equation", constant-linear-force-with-proportional-to-V-drag, etc.  That's
about where every text I've ever read stops, and with words to the effect
of, "Clearly this quickly gets too complicated to deal with...", introduces
the section on Euler integration.

But how much further can we actually go?  Can we add constant angular
velocity and/or acceleration and still get a closed solution?  Torque,
constant or otherwise?  Even, dare I say it, simple multibody interactions
(e.g. a hinge)?

What I'm getting at here (if you haven't guessed already) is more exact
solutions == fewer iterations + better system stability.  How far has this
tree been barked up?

-- 
Gary R. Van Sickle
Brewer.  Patriot.