I am working on a problem involving a particle sliding frictionlessly
on a particular shape of surface. An analysis using Newton's second
law is very messy because it is complicated to write down the normal
force. A Lagrangian analysis is straightforward and gives me a
second-order D.E. I can solve numerically. But suppose I wanted to
tackle this problem with students who haven't had Lagrange's equation
yet.
Well, I can use energy conservation. BUT... I now have a first-order
differential equation for the SQUARE of the speed. I can't see any
way to tell Maple (or whatever your favorite mathematical software
package may be) how to choose the correct sign for the square root in
each segment of the motion.
Physically I imagine I step the solution forward until I reach a
turning point and then I reverse the sign of velocity. But surely
there must be some way to instruct Maple to do this.
To simplify the problem, imagine throwing a ball straight up and it
then returns to your hand. I can write down F=ma, integrate once to
get v(t), and again to get y(t). All clear. Alternatively I can
re-express dv/dt as dv/dy * v and integrate to get v(y). But v isn't
a single-valued function of y! There are two possible signs for v for
a given y! The sign information has somehow been lost when I used the
chain rule. How can I recover it *given* that my goal is to find y(t)
starting from an equation for (dy/dt)^2?
I'm wondering how I've *ever* managed to solve mechanics problems
using energy. Clearly I'm somehow inserting the correct sign
information by hand, perhaps without even realizing I'm doing it. If
I can do it, there should be a way to tell Maple how to do it,
shouldn't there?