Talk:Generalized coordinates

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[edit] Subscripts

Erm, I fixed a mistake in the subscripts, but now I am not so sure there aren't bigger mistakes in the transformation equations for the double pendulum.

If we assume that the origin of the coordinate system is at the origin of the first pendulum, then wouldn't the transformation equations have to be

x_1=l_1*Sin(Theta_1) y_1=-l_1*Cos(Theta_1)

x_2=l_1*Sin(Theta_1)+l_2*Sin(Theta_2) y_2=-l_1*Cos(Theta_1)-l_2*Cos(Theta_2)

I am just sort of passing throught, so I don't really want to create an account to make these somewhat major changes.—The preceding unsigned comment was added by 216.211.78.195 (talk • contribs) .

I think you are right. I've made those changes, and I hope someone will look over them. Tom Harrison Talk 19:04, 25 May 2006 (UTC)

[edit] Opening Characterizations Don't Seem Right

As far as I can tell from general background/experience -- and a quick online search -- the term 'generalized coordinate' is not broadly used beyond a physics context where, as the article explains, it arises primarily from Lagrangian mechanics. Consequently, I doubt that it is correct to state that "Generalized coordinates indlude any nonstandard coordinate system applied to...". Let me elaborate.

  • First off the term "nonstandard coordinate system" is not defined here -- nor I suspect, anywhere.
  • Secondly, even if we stipulate that, for example, the Cartesian system and any coordinate system related to it by a linear transform constitutes the "standard" systems, we still have cases of parameterized coordinates which might not have anything to do with generalized coordinates in a Lagrangian problem.
  • Thirdly, what we are really talking about when we use generalized coordinates is a configuration space whose relation to an underlying real space is definable most of the time. It could even be the case that the geometry of the configuration makes it mimic a "standard" coordinate system.

My recommendation is that we drop the current opening characterization and focus instead on the fact that generalized coordinates reflect the configuration of the mechanical parts of the system per se and that each generalized coordinate could be a conventional coordinate, a position along a curve (i.e. a parameter), an angle (which in polar, spherical etc systems might be a conventional angular coordinate) or even some other dimensionless measure of the configuration.

I also think that we should early on introduce the fact that the dimensionality of the configuration space corresponds to the degrees of freedom of the problem (or can be so reduced) since "degrees of freedom" is an essential and important concept that comes into play whenever we use Lagrangian mechanics (it has an analagous role in statistics and I think a few other things).

I am not just jumping in to do any of these things because I think it starts a process of revision for this article that is serious enough to warrant discussion before implementation. And yes, I am implying that the whole article, though a nice start, is not appropriately robust and needs both addtions and revisions.

Which reminds me.. I value my own copies of Schaums Outline Series, but I would hesitate to cite any of them as authority. Its a pain to hunt down authoritative references, but it probably needs to be done here. Just citing Goldstein would be a suitable substitute.

Thanks --scanyon 21:21, 31 May 2007 (UTC)