>Three body problem such as the interaction between say, the sun, Earth and
>Jupiter. At the end of the nineteenth century Poincare' showed
>mathematically that the positions cannot be precisely predicted.
The three body problem is interesting becuase it is impossible to solve the equations of motion and predict where each body will be in the future. But this isn't the same thing as saying we don't understand the physics involved. The forces acting on each body at any point in time can be precisely determined. Since it is a non-linear problem, which doesn't have a nice closed form solution, you would have to rely on brute force computation to solve the problem. But since the final answer is sensitive to the exact conditions at every point in time (location, velocity, etc.), you would need an infinite amount of computation power to solve the problem this way. It is a practical, not a theoretical, difficulty.
>The solar system is not like a piece of clockwork however much the early
>modern scientists, mixing observation with mediaeval idealism about perfect
>spheres, hoped this was so.
You can construct very simple mechanical machines which behave like the heart - acting chaotically and "flipping" states like you mentioned in your earlier message. Something as simple as the equation Xn+1 = A*Xn(1-Xn), which is completely deterministic, can lead to "chaotic" behavior. This equation might be used to determine the deer population in a given forest for year n, where Xn is the normalized population (a fraction between 0 and 1).
Even assuming that this formula will EXACTLY predict the next year's population given this year's population, unless you have an exact count of the deer population this year your predictions may be worthless 20 years out, because this equation can be sensitive to initial conditions (for the right values of the constant A).
Systems which are completely deterministic can be unpredictable, and do weird things. That doesn't make them any less mechanical or deterministic.
Brett