Re: [Phys-l] solving an energy equation - revisited

[Jack Uretsky <jlu@hep.anl.gov>]

Hi all-
	Karl wrote:

> > 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?
> >
> > My brain is knotted up. Help! Carl
> > --
> > Carl E. Mungan,

	This is just geometry (and functional analysis).  You're dealing 
with a function that has two branches.  The same thing occurs if you 
consider integrating the differential equation for a circle:
	d/dx(y^2) =-2x
there is a solution for positive y and a solution for negative y.  The two 
solutions together map the whole circle.
	Since the solution is y = sqrt{const - x^2} it is obvious from the
square root that my solution must have two distinct branches, which
immediately tells me something about the physics.  In your case that
something is: going up ain't the same as going down.
						Regards,
							Jack

--
"Trust me.  I have a lot of experience at this."
		General Custer's unremembered message to his men,
		just before leading them into the Little Big Horn Valley