Talk:Lagrangian mechanics/Archive 1

message: 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.
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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
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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)
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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)

message: 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)
thread_title: generalized coordinates

message: 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)
thread_title:  gen. coord. added 

message: 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)
thread_title:  overloaded symbols 

message: 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)
thread_title:  stationary vs. minimum action integral 

message: 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)
thread_title:  New/old lagrange equations 

message: 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)
thread_title:  Lagrange multipliers and Cartesian Coordinates 

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

message: Hello everyone,
I am a bit confused about that generalised force derivation, Qj, 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)
thread_title:  Scalar potential vs. potential energy 

message: ::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♠nSoul 06:10, 24 February 2011 (UTC)
thread_title:  Proposed change, at another editor's request. Requesting feedback 

message: 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)
thread_title:  Removed last sentence from the intro 

message: 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)
thread_title: Clean up

message: 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) ⇄ ∑ici 07:49, 16 April 2012 (UTC)
thread_title:  Slight re-write 

message: 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) ⇄ ∑ici 16:40, 18 April 2012 (UTC)
thread_title:  Kinetic energy relations 

message: 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)
thread_title: Lagrange's eqns of the 1st kind

message: 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) ⇄ ∑ici 12:59, 21 April 2012 (UTC)
thread_title:  ''L'' for Lagrangian, script for density 

message: 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)
replies: text: 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)
thread_title:  Confusing example (is it wrong?) 

message: 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)
replies: text: Fixed. M∧Ŝc2ħεИτlk 15:25, 10 March 2013 (UTC)
thread_title:  Should the Lagrangian in the "2nd kind" box be in latin not script L? 

message: 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)
thread_title:  1.4 Lagrangian and action - issues 

message: 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∧Ŝc2ħεИτlk 09:55, 5 August 2015 (UTC)
thread_title:  Examples from [[Lagrangian]] 

message: #Lagrangian -> Lagrangian mechanics
#Lagrangian -> Lagrangian mechanics
#Lagrangian -> Lagrangian mechanics
#Lagrangian -> Lagrangian mechanics
#Lagrangian -> Lagrangian mechanics
#Lagrangian -> Lagrangian mechanics
So don't claim I stole anyone's work because the credit is given in these links! M∧Ŝc2ħεИτlk 11:26, 5 August 2015 (UTC)
thread_title: Summary of main content transfers

message: 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∧Ŝc2ħεИτlk 11:49, 5 August 2015 (UTC)
replies: text: 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∧Ŝc2ħεИτlk 21:49, 10 August 2015 (UTC)
thread_title:  Insane length 

message: I'm doing it now so please wait a while. M∧Ŝc2ħεИτlk 21:54, 10 August 2015 (UTC)
thread_title:  Cleaning up the references 

message: 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∧Ŝc2ħεИτlk 12:45, 21 September 2015 (UTC)
thread_title:  Transfer of the relativistic formulation to [[Relativistic Lagrangian mechanics]] 

message: 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∧Ŝc2ħεИτlk 14:43, 24 September 2015 (UTC)
replies: text: The quoted claim has been removed, the rest of the article needs tweaking. M∧Ŝc2ħεИτlk 14:15, 13 January 2016 (UTC)
thread_title:  Hamilton's principle and non-holonomic constraints 

message: 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∧Ŝc2ħεИτlk 09:10, 26 September 2015 (UTC)
thread_title:  Possible exchange of content 

message: Are Lagrange equations different from Euler-Lagrange equations?
Paranoidhuman (talk) 02:01, 12 May 2017 (UTC)
replies: text: No. "Lagrange equations", "Euler–Lagrange equations", and "Lagrange's equations" all mean the same thing AFAIK. JRSpriggs (talk) 06:55, 13 May 2017 (UTC)
thread_title:  Lagrange equations 

message: 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)
replies: text: This is just a guess. It might refer to  which I think (if I remember correctly) has that as a topic. JRSpriggs (talk) 21:27, 19 September 2017 (UTC)
thread_title:  Sloppy reference 

message: 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)
thread_title:  2.2 From Newtonian to Lagrangian mechanics - possible mistake? 

message: 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)
thread_title:  Notes and citations 

message: 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)
replies: text: 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)

text: 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)

text: (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)

text: 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)
thread_title:  Invariance under point transformations - misleading title