What I Think I've Done

Required: A mild knowledge of classical mechanics and perhaps some differential geometry.

So, assume you're working with a Hamiltonian H such that all solutions are regular curves in phase space. Let us limit our discussion to a 2 dimensional phase space, as that's easier to discuss. Now, it's fairly easy to show that each Frenet frame component is given by:

dt/ds * [a certain vector (for each component)]

The interesting thing that seems to pop up, however, is that these certain vectors are independent of the particular solution under discussion; i.e. they depend only on the point that one finds oneself at in phase space...for every solution curves passing through that point. Now, I don't think this seems immediately wrong, as dt/ds seems to carry within it solution-specific information, so that the uniqueness of each solution is no jeopardized. One consequence, however, is that two solution curves cannot 'cross' each other, i.e. if two solution curves meet, they would have to be tangent to each other.

Meh. Gotta run down to CVS.

EDIT: obviously, two solution curves wouldn't meet. Stupid me. And, clearly, the space would still be connected. Ah well, stupid me. Thanks, Keith.