Talk:Lagrangian mechanics

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Was redirected to Talk:Lagrange's equations.

Definitely needs a rewrite. A lot of the content here overlaps with the content in action (physics), but the derivation of the Euler-Lagrange equations differs. Some consolidation is probably in order, and I think I prefer the one here to the one in action (physics). There's definitely a notational issue, since this page uses r' and the other uses r-dot.

Taral 08:13, 19 Jun 2004 (UTC)

I have trouble understanding this bit below:

More generally, we can work with a set of generalized coordinates and their time derivatives, the generalized velocities: {qj, q′j}. r is related to the generalized coordinates by some transformation equation:

\mathbf{r} = \mathbf{r}(q_1 , q_2 , q_3, t). \,\!

What is q

What is this equation? \mathbf{r}(q_1 , q_2 , q_3, t). \,\!

The above equation makes no sense what so ever.

I cleared it up a bit, I hope, by reordering that sentence and adding a really simple example. Laura Scudder 00:08, 7 Mar 2005 (UTC)
Great article. But since the biggest clincher is generalized coordinates, perhaps there should be separate discussion on the matter elsewhere? --Rev Prez 13:10, 28 May 2005 (UTC)
"Generalized coordinates", according to Landau & Lifshitz, refers to any s quantities q1,q2,...,qs which completely define position in a system with s degrees of freedom. "Generalized velocities" are the associated velocities. You can think of them as vectors. For example, in a 3D system, which has three degrees of freedom, the usual way to think about the q variables are x, y and z. The Lagrangian works in spherical and cylindrical coordinate systems as well, which may be why the "generalized" label is used. - mako 30 June 2005 00:25 (UTC)
The generalized coordinates are really about picking the coordinates that make your life easiest. For instance, simple problems often have lower dimensional motion embedded in 3 dimensions, so you don't actually need 3 generalized coordinates. It all depends on whether the constraints on the motion are holonomic or not. So the generalized reminds you that your coordinates may need to be a totally non-traditional system, like the length along a wire bent into a weird shape (perhaps a bead is moving along the wire). --Laura Scudder | Talk 30 June 2005 01:16 (UTC)

Contents

[edit] generalized coordinates

The link generalized coordinates links to this page (Lagrange mechanics), I believe it would be nice to have either a larger discussion of generalized coordinates on this page or perhaps its own article. Article could give examples (such as how they are used in cartesian or spherical-polar coordinates) and discuss the relation to degrees of freedom. Perhaps a discussion on the related generalized momenta. Ideas? 71.131.37.89 02:37, 21 December 2005 (UTC)

[edit] gen. coord. added

Generalized coordinates now have their own article, which is not 100% yet, but is a good start. I'll continue to work on it and parse the content between this page and that over the next week.Jgates 03:13, 25 December 2005 (UTC)


  • As pointed before, nice article, however i find this lack on several "rare" but sometimes common examples involving Lagrangians:

L(q,q',q'',t) (Lagrangian with an acceleration term)

and the usual Hamiltonian Mechanics, i think in this case Hamiltonian is given by:

xp' + x'p'' − H(q,p,p') = L

these Hamiltonians happens in the euler-Lagrange equations for GR see "Ray D'inverno: Introducing Einstein's Relativity" (university course in Cosmology )

[edit] overloaded symbols

Symbols on this page like \mathbf{r}, \mathbf{v}, \mathbf{a} are overloaded to denote multiple functions. I realize this is a common practice is physics; stating which function a symbol refers to at key points, however, can make the article easier to follow.

Let's say A is the map that relates the coordinates qi, t to \mathbf{r}, ie \mathbf{r} = A(q_1, \dots, q_m, t), and fi is a function that gives an object i's coordinates qj, t with time t, \left\langle q_1, \dots, q_m, t \right\rangle = f_i(t). Frustrating ambiguities arise when the article discusses \mathbf{v}'s partial derivatives. In expression \mathbf{v}_i = \sum_{j=1}^m \frac {\partial \mathbf{r}_i}{\partial q_j} \dot{q}_j + \frac {\partial \mathbf{r}_i}{\partial t}, \mathbf{v}_i could be regarded as a function of only qj, t where \frac {\partial \mathbf {v}_i}{\partial q_j} has one meaning (and you're expected to assume the operator in \dot{q}_j commutes with \frac {\partial}{\partial q_j}) or \mathbf{v}_i could be regarded as a function of \dot{q}_j, qj (hidden in \mathbf{r}_i), t where \frac {\partial \mathbf {v}_i}{\partial q_j} has a completely different meaning or \mathbf{v}_i could be regarded as simply (A \circ f_i)^\prime = (A^\prime \circ f_i) f_i^\prime (the proper definition, a function of only 1 variable, t), among other possibilities. It seems to be the second one.

My point is the reader has to guess and shouldn't. --L0mars01 23:17, 7 November 2007 (UTC)

[edit] stationary vs. minimum action integral

While the action must be stationary for the entire path, Landau and Lifschitz argue that for sufficiently small displacements it needs to be a minimum (I think 2nd page of 3rd ed. ). I do not exactly see why this helps them, other than allowing them to prove that mass is positive (don't take my word for this though).. Could this possibly be worth a mention in the article? —Preceding unsigned comment added by 163.1.62.20 (talk) 13:55, 5 March 2008 (UTC)

[edit] New/old lagrange equations

What is the difference between the sections "lagranges equations" and "old lagranges equations"? Has it just been rewritten? Then the old one should be removed. If they have independently valuable content, then perhaps the titles should be changed, or at least a summary explaining what the differences between these lengthy derivations are. Feyrauth (talk) 01:35, 23 April 2008 (UTC)

[edit] Lagrange multipliers and Cartesian Coordinates

The aspect of (undetermined) Lagrange multipliers for the Lagrangian mechanics is missing in this article. Stressing this aspect would lead how to deal with systems witch still contains constraints e.g. using cartesian coordinates. I think this could be a start: http://electron6.phys.utk.edu/phys594/Tools/mechanics/summary/lagrangian/lagrangian.htm —Preceding unsigned comment added by 217.229.106.204 (talk) 08:57, 4 May 2008 (UTC)