Reduced to Quadratures

[Pete]

I'm unofficially taking a course in the gradute Classical Mechanics this semester. The text we are using is

 

Classical Mechanics - Third Edition, by Goldstein, Poole and Safko (2002. I'm trying to solve Exercise 17 in Chapter 2. The problem reads

 

 

It sometimes occurs that the generalized coordinates appear seperately in the kinetic energy and the potential energy in such a manner that T and V may be written in the form

 

[math]T = \Sigma f_i(q_i)(dq_i/dt)^2[/math] and [math]V = \Sigma V_i(q_i)[/math]

 

Show that Lagrange's equations then seperate, and that the problem can always be reduced to quadratures.

Where reduced to quadratures means expressed in terms of definite integrals which can be evaluated analytically or numerically. I can't see how to do that. I'll post the differential equation that results when one uses that Lagranian L = T - V later tonight. Thanks in advance.

 

Pete

[salaw]

Pete doesn't seem to have followed up with the diff eq's he was going to post, and since he also failed to mention something significant about the assignment I'll chime in here.

 

The thing he didn't mention is that the homework assignment here is "past tense" -- he handed it in last week, sans reduction to quadratures, and beyond saying more work should have been done, the prof neglected to provide a solution. So Pete is left scratching his head over the solution, as I am, since Pete talked to me about it, and I didn't see it either (uh, duh...). In any case this isn't "homework help" being requested here, at least not in the usual sense. At this point a complete solution would be much appreciated and would not be "cheating" in any sense of the word ... and so this is posted in general physics rather than the homework-help section.

 

Anyhow if we start with

 

[math]

{{\partial L}\over{\partial q_i}} =

{{d}\over{dt}}{{\partial L}\over{\partial \dot q_i}}

[/math]

 

where [math]L = T - V[/math], then, starting with the defs Pete gave (above, previous post) for T and V, and after taking derivatives and fiddling a little we get to

 

[math]

f'_i (q_i)\dot q_i^2 + 2 f_i(q_i) \ddot q_i + V'_i (q_i) = 0

[/math]

 

Of course the equations have separated, for sure, as each equation has just one coordinate in it. Next step is to "reduce it to quadratures" which apparently means rewriting it in terms of integrals of elementary functions.

 

Now, I am no differential equation whizz, to say the least, but this looks to me like a non-linear second order equation and I am pretty seriously unclear on how to get to *any* kind of solution from here, short of sticking the thing on a computer and solving it numerically (which we can't do, of course, since f is unspecified).

 

Goldstein says it can be reduced to quadratures, or more precisely he asks the student to show that it can always be reduced to quadratures (not exactly the same thing, eh?).

 

If anyone has a clue how to proceed, Pete would appreciate it, and since I got sucked into this, I'd appreciate it too.

 

Thanks in advance...

[Pete]

Thanks salaw. I was a bit off my game yesterday and didn't get around to uploading it and posting the link. I just did so. Its now at

 

http://www.geocities.com/pmb_phy/exercise_2_17.doc

 

I should note that one of the authors is an aqauntance of mine. I E-mailed him and he told me that I got the following response

Peter;

Your solution is correct and very detailed. Since each variable has its own differential equation the problem can be solved either formally or numerically. See http://mathworld.wolfram.com/Quadrature.html

 

So I'm scratching my head too. And I don't have extra hair up there to spare so please help soon!

 

Thanks salaw. I was a bit off my game yesterday and didn't get around to uploading it and posting the link. I just did so. Its now at

 

http://www.geocities.com/pmb_phy/exercise_2_17.doc

 

I should note that one of the authors is an aqauntance of mine. I E-mailed him and he told me that I got the following response

Peter;

Your solution is correct and very detailed. Since each variable has its own differential equation the problem can be solved either formally or numerically. See http://mathworld.wolfram.com/Quadrature.html

 

So I'm scratching my head too. And I don't have extra hair up there to spare so please help soon!