Casey

It's not always as simple as a pendulum, though. How could I analyze a more complicated constraint? An example I can think of is motion along a fixed path under the influence of 3 charges (kind of like the 3-body problem). Is there a general method that could work for analyzing all sorts of constraints?

The equation of motion for a pendulum is y''=-(g/L)sin(y), where y represents the angle of the stiff rod with respect to the vertical, g is the acceleration of gravity, L is the length of the rod. y also varies with respect to time. I've seen how this particular equation is derived, but I would like to know more about the general problem of modeling mechanical systems with constraints. Does anyone have some insight to offer? How could I view this problem from the mechanical perspective (finding positions, energies, momentums, etc.).

It would look like a "parabolic wave". It would be periodic with respect to time due to the lack of damping in the system. If you're looking for a "nice" equation, you're out of luck. However, I found a link about chaos theory and bouncing balls. It may not solve your problem, but it looks very cool: http://chaos.phy.ohiou.edu/~thomas/chaos/bouncing_ball.html. At the end points (when it hits the ground), it would be an in spontaneous change in motion just like the graph of the absolute value function. The magnitude of velocity would be the same just before and just after impact. In a computer algorithm, you could iterate the regular equation of motion over and over. In the algorithm, start on the ground going up (from the origin). The max of the parabola is the initial height from which the ball was dropped, so this is where the actual physical process starts. When the ball hits the ground, there will be a zero in the graph. Now, the second parabola will begin. It will have the same height and "spread" as the original. Start the new parabola from the zero that the original ended at and calculate its path to the next zero. Repeat ad infinitum. I hope this helps. If you've ever studied NKS, you'll be familiar with the limitations of mathematical models and the superiority of computer algorithms (that was a hint. Check out NKS!). Based on absolutely nothing except intuition, I think the path could be approximated by the absolute value of an elliptic function, which is a more general version of the trig functions we're all used to. I imagine this would be very difficult, though.

Hi. I'm Casey, and I'm just a sad shell of a former man (I discovered mathematics). I want to know how to do things without "doing" them (I'm too lazy for regular physics).