Talk:Generalized coordinates

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)

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)

I agree with you here. I came here looking for a good definition, and I didn't find it. What seems clear to me about generalized coordinates, are that they abstract away the choice of basis. What is not clear to me is what the span of the basis is. That is, are generalized coordinates limited to spatial configuration, spatial/angular configuration, or can other non-dimensional parameters enter into the basis (normalized mass values, for example). I suspect the answer is essentially spatial, but by Jacobian magic, that can be equivalent to angular configuration. Not sure what the directionality of time does to the concept of configuration space. 70.247.160.87 (talk) 04:55, 3 April 2013 (UTC)

Diagrams

Added diagrams to the article - hope they're clear eneogh.--Maschen (talk) 10:25, 22 March 2012 (UTC)

Diagrams II

Some of the diagrams on this page don't seem very professional (while others are excellent). Could we have a bit more consistency? If there is no response I will redo all the diagrams in Inkscape and upload as vector images.

Diagrams III

The diagrams explaining the generalized coordinates for one degree of freedom seem wrong for several of the coordinate examples: The angle and distance along an axis for the squiggly line are NOT bijective. Could somebody redo them with a less squiggly line that doesn't go back on its track? — Preceding unsigned comment added by 92.225.30.51 (talk) 22:56, 24 August 2013 (UTC)

Yes, these I meant to fix these. It is simpler to draw new ones and just delete the old one (File:Generalized coordinates 1df.svg). See my talk page here. Best, M∧Ŝc²ħεИ_τlk 22:18, 3 December 2014 (UTC)

Recent revisions

I have made a number of recent revisions in an attempt to explain why these coordinates are considered to be generalized versions of Cartesian coordinates. This article has quite a number of vague statements that do not seem helpful. I am trying to add some precision that I hope is more useful. Thank you for your patience.Prof McCarthy (talk) 00:48, 21 November 2012 (UTC)

Article rating

It seems that this article meets the criteria for a B-class article. Here is the description from Wikipedia:CLASSES: "A few aspects of content and style need to be addressed. Expert knowledge may be needed. The inclusion of supporting materials should also be considered if practical, and the article checked for general compliance with the Manual of Style and related style guidelines." I think this accurately describes the current state of this article. Prof McCarthy (talk) 21:11, 30 December 2012 (UTC)

Still needs rewriting

This article is about generalized coordinates, a simple topic. Yes, one can expect Lagrangian mechanics to be discussed, but solving problems with Lagrange's equations and D'Alembert's principle seems to be overdoing it.

I intend to rewrite bits of this article to expand more on the notion of generalized coordinates, quantities in terms of generalized coordinates, (velocity and acceleration vectors, kinetic energy), canonical momentum (which is not always the same as generalized momentum). I'll redraw some of my own diagrams, and will redraw the hideous pendulum diagram. M∧Ŝc²ħεИ_τlk 18:39, 27 August 2015 (UTC)

Move examples to Lagrangian mechanics?

Most of the examples (single and double pendulum) would fit better in the Lagrangian mechanics article, which involve the EL equations and the Lagrange equations using multipliers, and the virtual work examples could be moved to the virtual work article. For this article on generalized coordinates, the examples should be more qualitative, by saying what the generalized coordinates are, rather than "how to do Lagrangian mechanics with them", or "how to calculate virtual work with them". Of course virtual displacements can be mentioned, and the comparison between the virtual displacement in the full set of position coordinates (not independent according to holonomic constraints) and again for generalized coordinates. Any objections? Otherwise I'll transfer the content. M∧Ŝc²ħεИ_τlk 20:34, 27 August 2015 (UTC)

It's been longer than a month and no-one has objected, so I intend to do this later today or possibly tomorrow. The possible exchange of content has also been written a week ago here. M∧Ŝc²ħεИ_τlk 14:06, 2 October 2015 (UTC)

Please don't move. The examples seem to be showing how to use generalized coordinates. At any rate, each of the examples already have full-length articles devoted to them; they seem to be suitable here precisely because of the focus on generalized coordinates.67.198.37.17 (talk) 05:30, 14 January 2019 (UTC)

Genius

Notice that each paragraph is one sentence each. Wikipedia is shit.