Talk:Lagrangian mechanics/Archive 1

Archive 1

Would like to rewrite. The current article follows the historical development of Lagrangian mechanics which isn't the order which is most useful for teaching purposes. Also there should probably be links to example problems.

Wikipedia is going to get really cool once MIT puts more of their Open Courseware online.

A rewrite would be good. I wrote the article while going through Goldstein, thus the lengthy derivations which aren't very useful and in fact hurts the article. Also, go ahead and tinker with Hamiltonian, which maybe needs to be split up into Hamiltonian mechanics for classical physics and Hamiltonian for quantum mechanics. Good luck. -- CYD

A lot of TeX is needed here.

Be bold. -- CYD

Why does it say Lagrangian mechanics was demonstrated by Lagrange? Wouldn't "introduced" be a more appropriate word? "Demonstrated", in reference to theorems or other propositions, means proved. It is not said of whole theories, but of propositions. Michael Hardy 01:23 Mar 19, 2003 (UTC)

Does it make sense to have the euler-lagrange equation redirect to this page? We have a whole range of pages covering langragian, hamiltonian, langrangian mechanics, hamiltonian mechanics, and the calculus of variations. I'm not sure what the best way is, but I think something should be done to clean it up, and make it clear what is what. --Cronian 04:51, 17 May 2004 (UTC)

Unsorted text

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)

Yes I agree. Besides the issues of content differing etc, the introductory, defining statement is very questionable. It states that Lagrangian Mechanics is based on Hamilton's Principle, which is rather difficult to explain without time machines, or conceding that the term is badly named. This is simple confusion.

Lagrange died in 1813, presumably having completed the bulk of his work before that time. Hamilton was born in 1805, and was precocious, but not *that* precocious. — Preceding unsigned comment added by Stevan White (talk • contribs) 16:17, 2 July 2012 (UTC)

Stevan White (talk) 18:21, 20 March 2012 (UTC)

Kinetic energy relations

We are just about to calcuate a derivative of kinetic energy, I just wondered if it wouldn't be easier to put this part in a following way, without the need of further explaining "vanishing" 1/2 factor and so on:

{\frac {\partial T}{\partial {\dot {q}}_{j}}}={\frac {1}{2}}\cdot {\frac {\partial \sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}}{\partial {\dot {q}}_{i}}}={\frac {1}{2}}\cdot {\frac {\partial \sum _{i=1}^{n}m_{i}\mathbf {v} _{i}^{2}}{\partial {\dot {q}}_{i}}}={\frac {1}{2}}\cdot \sum _{i=1}^{n}{\frac {\partial m_{i}\mathbf {v} _{i}^{2}}{\partial {\dot {q}}_{i}}}

Now, according to $\,{{\frac {\partial }{\partial x}}f^{2}(x)=2f(x){\frac {\partial }{\partial x}}f(x)}\,$ , we have:

{\frac {\partial T}{\partial {\dot {q}}_{j}}}={\frac {1}{2}}\cdot \sum _{i=1}^{n}2\cdot m_{i}\mathbf {v} _{i}{\frac {\partial \mathbf {v} _{i}}{\partial {\dot {q}}_{i}}}=\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}{\frac {\partial \mathbf {v} _{i}}{\partial {\dot {q}}_{i}}}

Ender2101 10:50, 14 Feb 2009 (CET)

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: {q_j, 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

q_{1},q_{2},...,q_{s}

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)

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)

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 )

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 $q_{i}$ , $t$ to $\mathbf {r}$ , ie $\mathbf {r} =A(q_{1},\dots ,q_{m},t)$ , and $f_{i}$ is a function that gives an object $i$ 's coordinates $q_{j}$ , $t$ 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 $q_{j}$ , $t$ 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 ${\dot {q}}_{j}$ , $q_{j}$ (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)

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)

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)

Yes, why are two sections to derive the same equations? 189.179.243.82 (talk) 00:53, 16 January 2011 (UTC)

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)

Radical rewrite or a new article to the many already existing about more or less the same subject?

The basic form of the Lagrangian equations of motion valid also for non-conservative forces is:

{\dot {\overbrace {\frac {\partial T}{\partial {\dot {q_{i}}}}} }}=Q_{i}+{\frac {\partial T}{\partial q_{i}}}

where

T=T(q_{1},...,q_{n},{\dot {q_{1}}},...,{\dot {q_{n}}},t)

is the kinetic energy expressed in generalized coordinates

q_{i}=q_{i}(x_{1},...,x_{n},t)

where $x_{1},...,x_{n}$ the cartesian coordinates of the many mass points involved and $t$ is time

and

Q_{i}=\sum _{j=1}^{n}F_{j}\cdot {\frac {\partial x_{j}}{\partial q_{i}}}

are the generalized force components.

This relation is in my opinion easiest and most understandably derived by straightforward variable transformation (i.e. invariance under transformation) without reference to any variation principle!

If the forces are conservative, i.e

F_{j}=-{\frac {\partial V}{\partial x_{j}}}

one gets that

Q_{i}=-\sum _{j=1}^{n}{\frac {\partial V}{\partial x_{j}}}\cdot {\frac {\partial x_{j}}{\partial q_{i}}}=-{\frac {\partial V}{\partial q_{i}}}

and

{\dot {\overbrace {\frac {\partial T}{\partial {\dot {q_{i}}}}} }}={\frac {\partial T}{\partial q_{i}}}-{\frac {\partial V}{\partial q_{i}}}

One should also point out that ${\frac {\partial T}{\partial {\dot {q_{i}}}}}$ are the generalized momenta

p_{i}

and that the first order differential equation system to integrate is

{\dot {p_{i}}}={\frac {\partial T}{\partial q_{i}}}+Q_{i}

{\dot {q_{i}}}=\operatorname {F_{i}} (p_{1},...,p_{n},q_{1},...,q_{n})

where $F_{i}$ are the "inverse functions" to $p_{i}={\frac {\partial T}{\partial {\dot {q_{i}}}}}$ obtained by solving for ${\dot {q_{i}}}$

And examples how this differential equation system is derived, i.e. some concrete examples of explicitly derived functions

{\frac {\partial T}{\partial q_{i}}}+Q_{i}

and

\operatorname {F_{i}} (p_{1},...,p_{n},q_{1},...,q_{n})

should be included, possibly also with the result of a numerical integration of this system of first order differential equations

This would make the subject more praxis oriented and of use for a larger audience!

Stamcose (talk) 12:26, 17 July 2008 (UTC)

Scalar potential vs. potential energy

Hello everyone, I am a bit confused about that generalised force derivation, Q_j, in conservative field, given by scalar potential. The quantity V, this is evidently the potential energy, although the author denotes that as scalar potential and subsequantly use it as potential energy. Please, if there is anybody involved, explain it to me. Thanks a lot!

Pavol, Slovakia —Preceding unsigned comment added by 147.175.85.38 (talk) 19:41, 5 November 2009 (UTC)

Proposed change, at another editor's request. Requesting feedback

Hello, a new editor (User:Desfuscay) requested to make the following changes in this section. Since I don't know enough of the subject, can someone look it over and see if it is correct and okay to add?

indicating the presence of a constant of motion. Performing the same procedure for the variable $\theta$ yields:

{\frac {\mathrm {-d} }{\mathrm {d} t}}\left[m({\dot {x}}\ell \cos \theta +\ell ^{2}{\dot {\theta }})\right]+m({\dot {x}}\ell {\dot {\theta }}+g\ell )\sin \theta =0;

Thanks--Obsidi♠n^Soul 06:10, 24 February 2011 (UTC)

Removed last sentence from the intro

I removed the following sentence from the end of the intro: This does not appear to be true, since using standard Newtonian mechanics one would ignore an analysis of forces and use conservation of energy which follows directly from Newton's laws; alternatively one can eliminate the constraint force by resolving tangentially to the groove. It did not fit in and seemed like a topic of discussion. Hamsterlopithecus (talk) 11:20, 5 December 2011 (UTC)

Clean up

I made various changes as shown in the edit summary. I intend to clean up more but too busy for now. Also - what is "Old Lagrange's equations" supposed to mean when the result is identically stated in the article as the Euler-lagrange equations? Should no-one object "old" will be eventually removed. -- F = q(E + v × B) 13:45, 17 February 2012 (UTC)

Slight re-write

In the now called Derivation of Lagrange's equations (originally "Old Lagrange's equations"...), there are a few problems...

Why are there "6 degrees of freedom for 3 2nd order equations"? For each degree of freedom, there is one generalized coordinate, hence for N degrees of freedom there are N generalized coords. Then of course there are N generalized velocities and N generalized momenta. But the only things relevant to Lagrange's equations are the gen. coords and gen. velocities. Also, the first paragraph only mentions after the force-potential grad equation:

"Therefore, the motion of the particle can be completely described by 6 independent variables, or degrees of freedom."

This is wrong and/or misleading. What are the 6 independent variables/degrees of freedom? A force in 3d doesn't have to have 3 degrees of freedom, there could be less by the constraints on the system. "Generalized coords" are not synonymous with "generalized velocities" (are they? correct me if I'm wrong). They are treated as independent variables from gen. coords, but necessary for determining the motion of the system because the equations are second order (and are needed for generalized momenta in Hamiltonian mechanics). It’s not stated clearly here that:

"for N degrees of freedom, there are N corresponding generalized coordinates, and N corresponding 2nd order equations. Since the equations are 2nd-order, the generalized coords and corresponding generalized velocities determine the motion of the system for all times after the initial state. The generalized coordinates and velocities are treated independently, so there are 2N independent quantities that describe the motion of the system."

The pendulum example contradicts the statement - even if the gravitational force acting on the pendulum is conservative, there is still only one degree of freedom (usually taken as the angle), not 3 - and of course there is a corresponding generalized velocity also.

Using the force equation to talk about forces, then explaining gen. coords, right from the beginning does not help. It would be easier to introduce gen. coords in terms of constraint forces qualitatively, then give the derivation after.

Towards the end, the 2nd-last paragraph reads:

"The above derivation can be generalized to a system of N particles. There will be 6N generalized coordinates, related to the position coordinates by 3N transformation equations. In each of the 3N Lagrange equations, T is the total kinetic energy of the system, and V the total potential energy."

No - each particle has as many degrees of freedom as each needs, and again why 6 for each? Why are there 6N generlized coordinates but 3N transformation equations? Are there not as many transformation equations as there are for the original number of coordinates (say cartesian/spherical/whatever), which map to the generalized coordinates?

I'll re-write this section to be clearer, and break up the derivation slightly with headings so readers can have an overall qualitative summary of what's happening. Perhaps add a show/hide box also.

F = q(E+v×B) ⇄ ∑_ic_i 07:49, 16 April 2012 (UTC)

Kinetic energy relations

Sorry to remove so much of someone else's work - but the kinetic energy section was far too long and dense. Its been reduced to the minimum amount necessary, and hopefully more followable. In doing so - some sections have been moved around slightly, more logical ordering. F = q(E+v×B) ⇄ ∑_ic_i 16:40, 18 April 2012 (UTC)

Lagrange's eqns of the 1st kind

The presentation of Lagrange Equations of the First Kind follows H. Goldstein's book. Unfortunately, Goldstein is completely wrong. Take a look at this:

http://www.princeton.edu/~lam/documents/VirtualWork.pdf

S. H. Lam

Sauhailam (talk) 14:36, 30 August 2013 (UTC)

L for Lagrangian, script for density

The should be some pointer to the Lagrangian density to solve field problems, where the script L is used for Lagrangian density and normal Latin L for the usual Lagrangian. I changed all the scripts to Latin. It’s also typographically cleaner, nicer and easier to edit. F = q(E+v×B) ⇄ ∑_ic_i 12:59, 21 April 2012 (UTC)

Confusing example (is it wrong?)

I find the following confusing:

For example, for a simple pendulum of length ℓ, there is the constraint of the pendulum bob's suspension (rod/wire/string etc.). Rather than using x and y coordinates (which are coupled to each other in a constraint equation), a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation (and its time derivative) would be

$\mathbf {r} (\theta ,t)=(\ell \sin \theta ,t)\,\Rightarrow \,\mathbf {\dot {r}} ({\dot {\theta }},t)=(\ell \cos \theta ,t).$

which corresponds to the one degree of freedom the pendulum has.

Isn't r meant to represent the x and y coordinates at time t, that is, r(t)=(x(t),y(t))? We could also write r(θ)=(x(θ),y(θ)), which makes sense because θ=θ(t) also depends on t. Then it seems we'd have

\mathbf {r} (t)=\mathbf {r} (\theta )={\big (}x(\theta ),y(\theta ){\big )}={\big (}\ell \sin(\theta ),-\ell \cos(\theta ){\big )}

(where θ=θ(t)) and so

\mathbf {\dot {r}} (t)={\big (}\ell \,{\dot {\theta }}\cos(\theta ),\ell \,{\dot {\theta }}\sin(\theta ){\big )}

This looks rather different to $\mathbf {\dot {r}} ({\dot {\theta }},t)=(\ell \cos \theta ,t)$ above. Indeed, this doesn't seem to parse correctly, since the expression on the right depends on $\theta$ rather than ${\dot {\theta }}$ , as suggested by the left hand side. 109.77.20.41 (talk) 11:26, 7 May 2012 (UTC)

You're correct - it was unclear and partly wrong. It is fixed now. Thank you - that was an improvement. Maschen (talk) 11:44, 7 May 2012 (UTC)

Should the Lagrangian in the "2nd kind" box be in latin not script L?

Under Euler-Lagrange equations, there is a green box with "Lagrange's Equation (2nd kind)" on it. Inside this box, a script L instead of latin L is used for a lagrangian. Is it done on purpose or should it be latin as mentioned previously (here)? The examples that follow it uses latin L. Maybe I'm wrong. Weaktofu (talk) 22:58, 20 December 2012 (UTC)

Fixed. M∧Ŝc²ħεИ_τlk 15:25, 10 March 2013 (UTC)

Archive 1

1.4 Lagrangian and action - issues

I have three issues I'd like to see addressed.
1. The sentence "Technically[,] action is a functional, rather than a function: its value depends on the full Lagrangian function for all times between t1 and t2.' is confusing. Technically, a functional IS a function. I've yet to see a Wikipedia article using the term functional which explains WHY the use of that term should be preferred over function. All squares are rectangles. It is WRONG to claim a shape is a square "rather than" a rectangle. Virtually all readers are familiar with the term "function" from grade school, while few will be familiar with the term "funtional". This seems to me to be an attempt to use jargon without clear purpose. Consider this modification:"The action's value is a function (technically, a 'functional') of the full Lagrangian function for all times between t1 and t2." I believe the initial (confusing and mostly (as far as I can see) irrelevant) comment adds *nothing* to the sentence. What do others think? (If there is some purpose in distinguishing functions of the real numbers (or vector spaces of the same) from functionals of the vector space(s) of (analytic, differentiable,...) functions, then that purpose should be explained.)
2. If I recall my physics, it is quite possible that T and V, potential and kinetic energy, can not be separated for some systems. Meaning that the claim that L = T-V, while most often true, is not always true. If I'm right, then the statement that this equation DEFINES L is in error. (Was it disapative systems?, I no longer remember, sorry). So, I'm challenging this definition (although it is no doubt a common one, since most elementary applications will resolve into T and V terms.)
3."This also contains the dynamics of the system, and has deep theoretical implications (discussed below)." I have no idea where below the author is referring to. It would be helpful to be more clear and specific here.
4. Finally, the last paragraph (as of Dec 2,2014) of the next section (possibly the "below" referred to above?), claims:"However it is not widely stated that Hamilton's principle is a variational principle only with holonomic constraints, if we are dealing with nonholonomic systems then the variational principle should be replaced with one involving d'Alembert principle of virtual work. Working only with holonomic constraints is the price we have to pay for using an elegant variational formulation of mechanics." There is a huge problem with this! The article on holonomic constraints states:"a holonomic constraint depends only on the coordinates x_sub_j and time t. It does not depend on the velocities." This appears to contradict the entire substance of the Lagrangian Mechanics article, unless I've missed something?173.189.74.253 (talk) 21:29, 2 December 2014 (UTC)

Examples from Lagrangian

I transferred the examples in Lagrangian to this article, since they are relevant and useful here, and the field theory examples in that article to the (now new) Lagrangian field theory article. M∧Ŝc²ħεИ_τlk 09:55, 5 August 2015 (UTC)

Summary of main content transfers

So don't claim I stole anyone's work because the credit is given in these links! M∧Ŝc²ħεИ_τlk 11:26, 5 August 2015 (UTC)

Insane length

OK, finally done offloading stuff from Lagrangian. Now it remains to simplify and shorten this article (which shouldn't be too hard). I'll continue later today. M∧Ŝc²ħεИ_τlk 11:49, 5 August 2015 (UTC)

Well, now it is still very long because I have tried to explain the annoying terminology "holonomic" etc, define in this context what is meant by generalized coordinates, degrees of freedom, constraints, including easy-to-visualize examples, what the position and velocity vectors in terms of all this are, and finally D'Alembert's principle. In previous versions of the article the article put none of these things into context, saying clearly what these concepts are why and they are used. If I trim the section more it will remove concepts and terminology which should be in this article somewhere, because someone new to the topic will not know about them and will need context. M∧Ŝc²ħεИ_τlk 21:49, 10 August 2015 (UTC)

Recent add to lead

This,

It is not as general as the principle of stationary action because it is restricted to equilibrium problems.[ref deleted],

was recently added to the lead. Even if true (it is very true in some senses, find the right Lagrangian and you can derive anything you wish using the principle of stationary action), it serves mostly to confuse here imo. It is not clear what is meant by equlibrium problems. Moreover, D'Alembert's principle (at least in mechanics) is stronger yet. YohanN7 (talk) 20:17, 6 August 2015 (UTC)

Agreed. (I didn't add the statement btw). The entire article needs to be trimmed and clarified. "Equilibrium problems" sounds like statics. M∧Ŝc²ħεИ_τlk 21:01, 6 August 2015 (UTC)

I will take the liberty to remove it from the lead, it doesn't help and only lengthens the lead which is already long. The statement was trimmed to:

"The theory connects with the principle of stationary action,^[1] although Lagrangian mechanics is less general because it is restricted to equilibrium problems.^[2]"

and this will be removed. It can inserted back somehow, maybe we could start a "limitations" section summarizing the drawbacks of Lagrangian mechanics. M ∧Ŝc²ħεИ_τlk 16:53, 10 January 2016 (UTC)

For reference this edit is the edit in question. I have notified the editor. M ∧Ŝc²ħεИ_τlk 17:05, 10 January 2016 (UTC)

Reflist catcher

Cleaning up the references

I'm doing it now so please wait a while. M∧Ŝc²ħεИ_τlk 21:54, 10 August 2015 (UTC)

Better examples

The article finally has an improved ordering of topics, and conditions on potentials and energy conservation are clarified. It remains to trim the unfortunate walls of text (but the content on constraints and degrees of freedom is important to include in this article, for the reader's reference), and add much more interesting examples: Landau and Lifshitz has loads including multiple coupled harmonic oscillators, anharmonic oscillators, driven-forced oscillators, rigid bodies,... Hand and Finch also have a bunch of similar examples. I intend to add them later. M∧Ŝc²ħεИ_τlk 00:02, 16 August 2015 (UTC)

Well, the article now seems to have a better ordering of topics, but I have ended up complicating a number of simple things on too much generality (but some generality early in the article is definitely needed, like the kinetic energy in generalized coordinates, because most people probably think of the kinetic energy as a homogeneous function of degree 2, when in general it is not). I will simplify by restricting 3d (or lower dimensions), consider most cases for one particle and generalize to many particles after, and streamline the text and notation more.

But the "pedanticness" of the validity of the results is very important to emphasize in detail (which equations are more or less general than others, energy conservation, the nature of potential energy for different forces, etc.). Also, the recently added topics like Newton's 2nd law in curvilinear coordinates, and the geodesic equation, are not irrelevant because they are exactly what is involved with Lagrangian mechanics, they do in fact provide the natural extension to general relativity, and citations have been added to support what has been said (even more references will be added soon).

The mentioned examples can be done last thing. Still rewriting... M∧Ŝc²ħεИ_τlk 18:22, 25 August 2015 (UTC)

Transfer of the relativistic formulation to Relativistic Lagrangian mechanics

In this edit I moved the relativistic formulation to an article worthy of its title. This article should concentrate on the non-relativistic formulation, and its too long to include relativistic and non-relativistic anyway. M∧Ŝc²ħεИ_τlk 12:45, 21 September 2015 (UTC)

Out of order

the sections seem to be badly out of order. The discussion of how the system handles constraint forces comes before the definition of what a Lagrangian is? Would a person who didn't already understand what a Lagrangian is understand the article? Seems to need a rewrite to have the introductory material first. Geoffrey.landis (talk) 03:16, 24 September 2015 (UTC)

I disagree the sections are "badly out of order".

First the introduction shows examples where Lagrangian mechanics is useful, then the general definitions of the position and velocity vectors, and constraint equations and generalized coordinates, follow immediately to show how the examples fit in the with general definitions. This is the "introductory material first".
Then the Lagrangian is defined along with the definitions of kinetic and potential energy, and in painful detail the terms "explicitly dependent on time" etc. are spelled out.
Then are the equations of motion, followed by the transition from Newtonian to Lagrangian mechanics, and in the process the origins and validity of the equations are shown (why they are true and where they come from).
Then there are properties of the EL equations relevant to mechanics.
Then two coordinate examples, and two more detailed mechanical examples.
Finally there are applications of the theory in other contexts.

So I would say this is very much in order, because there is motivation first thing for why and where the theory is even useful, and the definitions are introduced as needed and given in extremely explicit terms.

Do you think it is bad to motivate with examples first? I was suspecting people would complain for providing no motivational introduction, so put it first, followed by the general definitions of coordinates and velocities and constraints and the Lagrangian before the equations of motion. You don't need the definition of a Lagrangian to understand the examples, nor the ideas of constraints.

I'll wait for at least your reply before I try once more to rewrite the sections "in order". M∧Ŝc²ħεИ_τlk 09:09, 24 September 2015 (UTC)

I suppose we could do something like this (though I'm not keen on it):

compress all definitions of positions, velocities, generalized coordinates, constraint equations Lagrangian, and kinetic and potential energies with the equations of motion,
move the motivational examples in the introduction either to
- the examples section (after the coordinate examples, before the detailed examples), or
- at the end of the more detailed discussion of Newton's laws before D'Alembert's principle.

Does that sound better? M∧Ŝc²ħεИ_τlk 09:32, 24 September 2015 (UTC)

You're missing the salient point altogether, a problem that plagues the majority of article-writers who love their subject more than widest dispersion of its understanding:

"Would a person who didn't already understand what a Lagrangian is understand the article?"

You need the exposition to follow a pedagogical order, rather than the order of most logical/elegant progress, an order best suited to, say, ice-dance judges at an Olympics event who hold up their marks. For the latter, you get 5/6, 6/6, 5/6; keep practicing and you'll get solid sixes... from mathematicians and post-grads. If that's your audience, go for it. JohndanR (talk) 20:03, 6 March 2021 (UTC)

Hamilton's principle and non-holonomic constraints

Someone (not me) has written:

It is not widely stated that Hamilton's principle is a variational principle only with holonomic constraints, if we are dealing with nonholonomic systems then the variational principle should be replaced with one involving d'Alembert principle of virtual work.

Actually, Hamilton's principle does work for (certain?) nonholonomic constraints, but things get more messy. We shouldn't go into too much depth in this article, but it should at least say Lagrange's equations of the first kind can be formulated with non-holonomic constraints, and make amendments elsewhere throughout where it says Lagrangian mechanics "only works for holonomic constraints". The main article (Hamilton's principle) should contain the formalism with holonomic and nonholonomic constraints (which it currently does not). As always, a good source for all this is Goldstein. I'll get back to this soon. M∧Ŝc²ħεИ_τlk 14:43, 24 September 2015 (UTC)

The quoted claim has been removed, the rest of the article needs tweaking. M ∧Ŝc²ħεИ_τlk 14:15, 13 January 2016 (UTC)

Possible exchange of content

As another possible remedy for shortening, enhancing, and better organizing this article (which may also enhance the generalized coordinates article in the process)...

The generalized coordinates article has examples which reflect better the applications of D'Alembert's principle and Lagrangian mechanics. This article has an introductory section motivating the use of generalized coordinates and how constraint equations are formulated, first with examples then generally (for holonomic constraints).

Maybe we could

move the content in Lagrangian mechanics#Introduction to somewhere in generalized coordinates,
move the examples in Generalized coordinates#Examples to the examples section in this article.

M∧Ŝc²ħεИ_τlk 09:10, 26 September 2015 (UTC)

Reverts in lead

Someone has removed [1][2] from the lead the correct statement that Lagrangian mechanics does not introduce new physics and is less general than Newtonian physics. It is explained why in the article. I have reverted both edits. Let's see what user:AHusain3141 has to say about this exercise 3.24 in Jose Saletan. M ∧Ŝc²ħεИ_τlk 22:12, 20 February 2016 (UTC)

I mention the exercise 3.24 of Jose and Saletan in the undo notification. That gives an example of how you can account for such forces in Lagrangian mechanics as well by using a judicious change of variables. So it is not really less general in the way described. The not introducing new physics statement is strictly true but not morally true because it does give the deformation theory for the quantization by the hessian at the critical point and not just the equation of motion (the critical point). AHusain3141 (talk) 22:21, 20 February 2016 (UTC)

Firstly, Lagrangian mechanics does not introduce new physics, does it? So your removal of that was wrong. Also is this Jose and Saletan a reliable source? Please explain how they do this "judicious change of variables". Can you actually account for dissipative forces exactly as Newton's laws can? M ∧Ŝc²ħεИ_τlk 22:23, 20 February 2016 (UTC)

I have found the book here and will investigate what I can. M ∧Ŝc²ħεИ_τlk 22:26, 20 February 2016 (UTC)

Looks like an impressive book and could be a valuable reference for this article. Looking at p.129 onwards, some of it agrees with what is already in the article, but we need to be careful. It seems the forces have to take certain forms to fit into the modified EL equations. I'll continue reading and maybe try to get a copy asap, thanks for pointing it out. M ∧Ŝc²ħεИ_τlk 22:38, 20 February 2016 (UTC)

In the meantime, feel free to rewrite the lead, but please do not remove the no new physics statement, it could be misleading.

Maybe you could say something like "Lagrangian mechanics can account for dissipative and driven forces by splitting the external the forces into a sum of potential and non-potential forces, leading to a set of modified EL equations", or similar. M ∧Ŝc²ħεИ_τlk 22:47, 20 February 2016 (UTC)

Jose and Saletan is a reliable source and a good book. The book's exercise 3.24 is an example of including a particular type of dissipation into Langrangian mechanics. Rayleigh's dissipation function is another way. So implying in the lead that Lagrangian is less general than Newtonian because Lagrangian cannot handle dissipation/non-conservative forces isn't really true. Can Lagrangian mechanics handle every kind of non-conservative force Newtonian mechanics does? I don't know. What does Feynman actually say about this? I agree that the Lagrangian doesn't add new physics as far a classical mechanics is concerned. If one wanted to point out the added utility of the Lagrangian in quantum or semiclassical mechanics, I think that would be OK, too. --Mark viking (talk) 23:01, 20 February 2016 (UTC)

Thanks for contributing here. The original claim and reference to Feynman was by user:Ywaz in this edit. I left it that way (and reworded to say "less general") because I thought it was true, that Lagrangian mechanics could handle some (not all) types of dissipative or driven forces, just not as broadly as Newtonian mechanics. To be on the safe side, I'll tweak the lead and add the reference. M ∧Ŝc²ħεИ_τlk 23:22, 20 February 2016 (UTC)

Yeah that looks good. Thanks for putting the ref. AHusain3141 (talk) 23:53, 20 February 2016 (UTC)

I agree, the reword looks good. Thanks, --Mark viking (talk) 01:13, 21 February 2016 (UTC)

It might be good to mention non-conservative forces are not fundamental (friction is not fundamentally real). Newton's law expressed as a negative of the potential gradient is a correction to Newton's original laws that makes them conservative. So the original newton's laws alone are not capable of deriving modern mechanics because conservation of energy (specifically heat) was not included. This is my interpretation of Feynman, but it might be that "equal and opposite reaction" strongly implies conservation of energy. The Feynman reference to Newton's law as non-fundamental (non-conservative forces) was in his least action chapter of the 3 intro physics books. The principle of stationary action can solve some problems lagrangian and hamiltonian can't. I believe it was "non-equilibrium" problems but I did not fully understand the paper. Ywaz (talk) 10:22, 21 February 2016 (UTC)

It isn't a question of whether the forces are fundamental or not. It is known Lagrangian mechanics can (at least sometimes) handle velocity-dependent forces, the EM field is the prototypical example.

What needs to be clarified is can Lagrangian mechanics handle any type of force that Newtonian mechanics can? When extensively rewriting, I have tried to be very careful in linking potential energy and forces (conservative, velocity-dependent, time-dependent), and have used the phrasing "if the force can be derived from a potential in this way".

Could you be specific on the Feynman "intro" books, are they just his 3-volume lectures of physics or something else? I'll check what's in his 3-volume lectures (volume 2 seems to be the relevant one). M ∧Ŝc²ħεИ_τlk 16:06, 21 February 2016 (UTC)

Euler-Lagrange equation represents an equilibrium of generalized forces, once it has been obtained for conservative forces, I guess that one can add to it whatever force he likes. But then the problem is: what is the Lagrangian and the action for such ′modified′ system? Can we prove that it satisfies the principle of stationary action? Are we still talking about Lagrangian mechanics? (No rhetorical questions, I don't know the answers)

In these Notes on Quantum Mechanics at page 5, the author obtains the action for a particle moving in an electromagnetic field. A velocity potential is added to the usual Lagrangian, that is (with different notation):

L(\mathbf {q} ,{\dot {\mathbf {q} }},t)=\cdots +\mathbf {a} (\mathbf {q} ,t)\cdot {\dot {\mathbf {q} }}

Such velocity potential renders then in the Euler Lagrange equation as:

\cdots +{\frac {\partial \mathbf {a} }{\partial t}}(\mathbf {q} ,t)+\left(\left({\frac {\partial \mathbf {a} }{\partial \mathbf {q} }}(\mathbf {q} ,t)\right)^{T}-{\frac {\partial \mathbf {a} }{\partial \mathbf {q} }}(\mathbf {q} ,t)\right){\dot {\mathbf {q} }}=\mathbf {0}

Note that we seem to end up with a damping matrix that is zero-diagonal symmetric. Esponenziale (talk) 01:44, 28 February 2016 (UTC)

To Esponenziale: The vector potential appears in Relativistic Lagrangian mechanics#Special relativistic test particle in an electromagnetic field rather than classical Lagrangian mechanics. JRSpriggs (talk) 03:07, 28 February 2016 (UTC)

Yeah, I didn't noticed that it appears also in this page as well (Lagrangian_mechanics#Electromagnetism). But what I was saying holds anyway: forces that have potential can be accounted in the Lagrangian (and therefore in the action), while it seems that any other force has to be introduced a posteriori as a generalized force. Consequently, if there're forces without potential, then the motion of the system isn't the one that renders the action stationary, but it is the one that satisfies the equilibrium of the generalized forces.

I guess that Rayleigh function importance is just that it allows to fully describe forces that don't have potential with a simple scalar-valued function. Therefore it seems that:

If all forces have potential, the mechanical system is fully described just by a scalar-valued function that is the Lagrangian,
Otherwise, the mechanical system may be fully described just by the Lagrangian and the Rayleigh function. Esponenziale (talk) 19:25, 28 February 2016 (UTC)

Newton's laws: Covariance/Contravariance mess-up

Newton's second law is given as

F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{a}}}-{\frac {\partial T}{\partial \xi ^{a}}}.

While the first parts of the equation imply that the force is a contravariant vector, the latter -- defining the force in terms of derivates of the kinetic energy -- defines the force as a covariant vector. We could simply split this into two equations

F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)

F_{a}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{a}}}-{\frac {\partial T}{\partial \xi ^{a}}}

but I'm afraid this would be quite confusing. Because the force is typically referred to as a covariant vector, maybe the whole equation should be rewritten into that form. Which one should we choose?

You need to use the metric tensor to lower the index of the velocity (contravariant) when calculating the momentum (covariant). See User:JRSpriggs/Force in general relativity.

Also see Relativistic Lagrangian mechanics#Lagrangian formulation in general relativity for a variation which leaves the momentum contravariant. JRSpriggs (talk) 18:57, 30 October 2016 (UTC)

Either which one would do, just as long as equations are consistent. Perhaps

F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ab}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{b}}}-{\frac {\partial T}{\partial \xi ^{b}}}\right)

is the quickest fix. YohanN7 (talk) 13:18, 31 October 2016 (UTC)

Sorry for late involvement. I actually think that splitting the equations may be better since many books just write force in terms of the derivatives of kinetic energy without the metric tensor. We can always link to raising and lowering indices if the reader wants to know how to do that. But combining the equations on the same line makes comparison easier (and saves space). I'll check the sources and try coming back to this tomorrow morning (but expect longer delay as always)... M ∧Ŝc²ħεИ_τlk 09:50, 3 November 2016 (UTC)

Lagrange equations

Are Lagrange equations different from Euler-Lagrange equations? Paranoidhuman (talk) 02:01, 12 May 2017 (UTC)

No. "Lagrange equations", "Euler–Lagrange equations", and "Lagrange's equations" all mean the same thing AFAIK. JRSpriggs (talk) 06:55, 13 May 2017 (UTC)

Sloppy reference

What is 'The Principle of Least Action, R. Feynman' in the references, a reference to? 31.50.156.42 (talk) 17:18, 19 September 2017 (UTC)

This is just a guess. It might refer to Feynman, Richard P. (1967). The Character of Physical Law: The 1964 Messenger Lectures. MIT Press. ISBN 0-262-56003-8. which I think (if I remember correctly) has that as a topic. JRSpriggs (talk) 21:27, 19 September 2017 (UTC)

2.2 From Newtonian to Lagrangian mechanics - possible mistake?

I am fairly sure this statement is false, specifically the first part- "Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero". The issue is that constraint forces aren't always perpendicular to displacement (consider a pair of particles constrained to be a fixed distance from one another; if a force is applied to one but not the other along their shared axis, a virtual displacement along that axis would be parallel to the constraint forces). Now it is true that the total work done by constraint forces over the entire system is zero, but this is not true on a per-particle basis.

While this mistake is small, it is important. I was hesitant to write this in the talk page because I'm fairly new to editing wikipedia, the statement had a citation, and I have been using this article to try and understand Lagrangian Mechanics and so obviously am not an expert on the subject (my uni courses were inadequate on the subject). But this mistake planted a misunderstanding in my brain about d'Alembert's principle which took me hours and hours of contemplating to recognise and fix. If it had such a big impact on me, it might be worth both fixing and perhaps doing further edits to pre-empt this mistake for future readers. — Preceding unsigned comment added by 121.222.27.228 (talk) 12:11, 13 May 2018 (UTC)

Notes and citations

Dear Editor, The bibliographic notes are very poor. I do not know who has edited them. I have verified that the links are not working at all. Please could anybody help to revise them so that they can point to the source? Linking the notes to the bibliographic sources may be preferring high-quality open-source references could be much more clear. Actually they appear as unworking links. Kind regards Enrico — Preceding unsigned comment added by 94.36.88.238 (talk)

~~Could you be more specific? Which links are dead?~~ Oh, never-mind, I see what you mean. The Harvard citation links have not been set up. I have fixed the first four. I will do some more soon. Each of the citations should be in the references section below. You can find them without the links. Richard-of-Earth (talk) 16:18, 15 April 2020 (UTC)

Invariance under point transformations - misleading title

@Ponor: You reverted my edit suggesting that I refer to this page. Problem is: I did read that page before doing the edit. And I do agree that invariance, on its own, is a broad term. However, that term is better than the title you are insisting on which is "invariance under point transformations". In the edit message, I explained exactly why I object to this term: no points as such are being transforms. Only coordinates are. The title you are insisting on is misleading. StrokeOfMidnight (talk) 21:53, 3 August 2020 (UTC)

Hey. It's just a traditional name for this kind of transformation, you cannot change that. So either leave it as it is, or call it "Invariance under coordinate transformation" ("Invariance under generalized coordinate transformation"). You have to be more specific, that's all. It's not "invariance under everything" as your title would imply. Cf invariance Ponor (talk) 23:27, 3 August 2020 (UTC)

PS I am going to change the title to "Invariance under coordinate transformations". StrokeOfMidnight (talk) 22:37, 3 August 2020 (UTC)

One additional note: your point transformation can be something like q=s·exp(ωt), so it's not that that you're only changing from, say, Cartesian to spherical coordinates, time can be involved too. "Point transformation" does not mean what you think it means, "points are not being transformed" does not really mean anything (how would you transform points?). Check https://solitaryroad.com/c282.html or google it. I still think the old title was better, but will leave it up to you. Cheers! Ponor (talk) 23:58, 3 August 2020 (UTC)

Well, a "point transformation" would send a point to another point, as in

(x,y)\mapsto (x+1,y+1).

Here, on the other hand, we aren't moving points around. All we are doing is change our view of them by switching to new coordinates, even if the switch itself is time-dependent. Mathematically speaking, if $M$ is the configuration space, and $TM$ the tangent bundle, then the invariance simply means that $L:TM\times \mathbb {R} _{t}\to \mathbb {R}$ is a well defined smooth function. Do we understand each other? Regards, StrokeOfMidnight (talk) 00:32, 4 August 2020 (UTC)

Well, you're trying to rewrite history. No one at wikipedia invented the name for this kind of transformation, and the name is still in the first sentence following your new title. So tell me, are these point transformations, q=s·exp(ωt) or q=t/(η+s), moving points around? (What are "points" to you?) Of course they are, that's the idea. Your (x,y) ↦ (x+1,y+1) is (x',y')=(x+1,y+1), and yes, your new generalized coordinates (x',y') would be "moved around". Your new description of the system would be in the new coordinates, and your Lagrangian eqs would keep the same form. Nothing prevents you from doing this transformation for two particles with two degrees of freedom each: (x1',y1', x2', y2')=(x1+y2,ωt,y1-x1,ωt) - so which exactly points (particles?) does the new set represent? Point transformation = a very special change of variables. "Points" do not get to keep their identity, so what's being "moved"? Ponor (talk) 01:14, 4 August 2020 (UTC)

Ok, for the sake of this article, I concede and am going to self-revert. But as far as points not being able to keep their identity: if you a mark a point on the board, you can refer to that point later using as many coordinate systems as you wish, and it'll still be the same point. Yes, the coordinates themselves are being moved around, but the point isn't. So, the point is not being transformed (or "moved"). Regards, StrokeOfMidnight (talk) 01:54, 4 August 2020 (UTC)

Examples of Use (Applications to CS and Engineering)

Would it be wise to include examples where Lagrangian Mechanics are useful compared to Newtonian Mechanics? CessnaMan1989 (talk) 03:23, 30 September 2020 (UTC)

Sure! We can discuss them here first, if you want. Which ones did you have in mind? Ponor (talk) 04:40, 30 September 2020 (UTC)

Ok, I'm sorry I had already put two in. I was thinking of mentioning how Lagrangian Mechanics are convenient for tracking drones, and I linked to a paper about it. I was also thinking of posting articles about how it has inspired many Deep Learning and AI algorithms too.CessnaMan1989 (talk) 20:45, 1 October 2020 (UTC)

@CessnaMan1989: with all due respect, I don't think your addition belongs to this article. Shouldn't this rather be an example of optimization problems or calculus of variations? The whole new section reads like an advertisement, and to an uninitiated doesn't really mean much. In the paper you cited (are you one of the authors? careful with that, wiki prefers secondary sources, not original research!), what would even be the F=ma equation, what's the recipe (I am asking because you're saying Lagrange is easier than Newton)? What are T and -V in the Lagrangian? How does one generalize to other similar optimization problems? Ponor (talk) 23:55, 1 October 2020 (UTC)

I agree with Ponor. Also, claims like the one CessnaMan1989 is making need to be backed up by several mutually independent sources. Otherwise, this is a WP:POV issue. StrokeOfMidnight (talk) 04:40, 2 October 2020 (UTC)

Okay, :@Ponor: and :@StrokeOfMidnight: I will revert the edit and draft before posting again. I'm definitely NOT one of the authors of this paper, but I was just offering it as an example. I can find lots of original research to cite, but only a few secondary sources on the matter simply because this is a very niche area of academia. Most people who learn about Lagrangian mechanics, especially the engineers, only do so in graduate school now. As a result, there just aren't as many textbooks on the subject. Still, there are textbooks such as "Theory and Applications of Multitethers in Space" by Panfeng Huang and Fan Zheng that demonstrate the benefits of Lagrangian Mechanics in simulations. — Preceding unsigned comment added by CessnaMan1989 (talk • contribs) 15:45, 4 October 2020 (UTC)

Curly L?

Would it not be more standard / look better to uses a curly ${\mathcal {L}}$ for the Lagrangian rather than a regular capital L?

Blitzer99 (talk) 22:21, 2 December 2020 (UTC)

@Blitzer99: I agree. This is a convention I've usually seen followed, especially because in a lot of places people use L for angular momentum as well, which may lead to confusion. It ought to be changed. EffyJohn (talk) 15:43, 30 September 2021 (UTC)

[1] Goldstien, Poole & Safko 2001, p. 35

[2] ttp://williamhoover.info/Scans1990s/1995-10.pdf

[1]

[2]

Fahrer / Team für die Suche eingeben:

Fahrer / Team für die Suche eingeben:

Langbahn Team – Weltmeisterschaft

Unsorted text

Kinetic energy relations

generalized coordinates

gen. coord. added

overloaded symbols

stationary vs. minimum action integral

New/old lagrange equations

Lagrange multipliers and Cartesian Coordinates

Radical rewrite or a new article to the many already existing about more or less the same subject?

Scalar potential vs. potential energy

Proposed change, at another editor's request. Requesting feedback

Removed last sentence from the intro

Clean up

Slight re-write

Kinetic energy relations

Lagrange's eqns of the 1st kind

L for Lagrangian, script for density

Confusing example (is it wrong?)

Should the Lagrangian in the "2nd kind" box be in latin not script L?

1.4 Lagrangian and action - issues

Examples from Lagrangian

Summary of main content transfers

Insane length

Recent add to lead

Reflist catcher

Cleaning up the references

Better examples

Transfer of the relativistic formulation to Relativistic Lagrangian mechanics

Out of order

Hamilton's principle and non-holonomic constraints

Possible exchange of content

Reverts in lead

Newton's laws: Covariance/Contravariance mess-up

Lagrange equations

Sloppy reference

2.2 From Newtonian to Lagrangian mechanics - possible mistake?

Notes and citations

Invariance under point transformations - misleading title

Examples of Use (Applications to CS and Engineering)

Curly L?