Re: [Phys-l] solving an energy equation

[John Denker <jsd@av8n.com>]

Carl Mungan 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.

There is a profound physics lesson here.  The energy (aka the Hamiltonian)
is fundamentally less informative than the Lagrangian.

If you tell me the Lagrangian (and tell me your choice of coordinate)
I can tell you the Hamiltonian ... but the converse does *not* hold:
if you give me the Hamiltonian (and the coordinate) I cannot in general
tell you the Lagrangian;  you would need to tell me what momentum-variable
is conjugate to your position-variable.

"The Lagrangian knows all and tells all."

> 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.

If I were timestepping the equations of motion, I would keep track
of two variables:
 -- the position, and
 -- the momentum.

Given (x,p) at a given time, I can use the equation of motion to calculate
(x,p) at a slightly later time ... and then iterate.  I need to keep two
variables.  This can be explained by the fact that the equation of motion is
a *second*-order differential equation.  Each step in the time-stepping process
is an initial-value problem, requiring two "initial" values.