Lagrangrian and Hamiltonian mechanics: A historical picture

[Omega0]

TL;DR Summary

It appears to me pretty clear where QM, SRT or GRT comes from, I would like to understand where the classical mechanics comes from. Was there a problem?

Hi,

I believe that I have an acceptable level of understanding where SRT, GRT, QM and QFT come from. This is not true for me regarding the "good old stuff".
Newton, okay, this is relatively () clear to me but do you know something about the historical motivation for Lagrangian and Hamiltonian mechanics? (Please note, I know the theory quite well, I am really speaking about the idea, the history)

If you would ask me what was the reason to develop it I would just guess in the dark and say something like "I believe that Lagrange had the motivation from seeing that just having time and x,y,z as coordinates is not well suited for problems where the real movement reduces to something easier. A better coordinate system could be used. So Lagrange came up and found something to describe systems easier. He came up with the Lagrangian function but this so-called action is not the energy of a system. Hamilton thought about it and found via a transformation a picture where the system energy is used instead of the action."

Well, just my story, just something which sounds quite logical to me but I am interested in the real history, the real motivation.

Please let me know if you have something for me.

Thanks!

 

[Frabjous]

This thread quotes Hamilton directly
https://www.physicsforums.com/threads/principle-of-stationary-action-intuition.1011392/#post-6596319

You might try looking at Lanczos
https://www.amazon.com/dp/0486650677/?tag=pfamazon01-20
Rojo and Bloch
https://www.amazon.com/dp/0521869021/?tag=pfamazon01-20
Coopersmith
https://www.amazon.com/dp/0198743041/?tag=pfamazon01-20

 

[Omega0]

This thread quotes Hamilton directly
https://www.physicsforums.com/threads/principle-of-stationary-action-intuition.1011392/#post-6596319

Awesome, hard to read but explains a lot.

Thanks.

 

[Cleonis]

About the things that motivated Joseph Louis Lagrange:

A translation of 'Mecanique Analytique' is available on Archive.org
'Mechanique Analytique'

The following is my interpretation
(As in: don't take my word for it, go to the original source):

The structure of 'Mecanique Analytique' is that Lagrange first sets out to cover a wide range of cases in Statics.

Let me give the simplest example (my example, I haven't looked up whether Lagrange discusses it): a marble in a bowl. The marble comes to rest at the lowest point in the bowl. For that case that is the equilibrium point.

Lagrange discusses how to set up a method that generalizes, so that for many different cases one can identify the equilibrium point. For instance, Daniel Bernoulli had proposed to give the object a small nudge (a virtual displacement), and to evaluate how much velocity the object will gain as gravity moves it towards the equilibrium point. The equilibrium point is the point where for all nudges you obtain the same result for expected velocity. In Lagrange's time several approaches like that were in circulation.

For problems in Statics Calculus of Variations is well suited. Lagrange is credited with further development of Calculus of Variations. Lagrange did the development work on Calculus of Variations for application in Statics. For evaluation of effect of virtual displacement I think Lagrange settled on evaluating change of energy.

That said, at the time the concept of 'energy' was not in the form that we know today.

It was known, of course, that in perfectly elastic collission a quantity proportional to $mv^2$ is conserved. That quantity was named 'vis viva', the 'living force'.

This 'vis viva' is of course a precursor to our modern notion of kinetic energy. The point is: at the time the concept of 'vis viva' was a standalone concept, it was not recognized as being part of a larger conserved quantity concept.

Lagrange had a concept corresponding to what today is referred to as potential energy, and he had the relation with vis viva, but he did not have that in the modern form.

That is: Lagrange did not have the modern form of the Work-Energy theorem.

(Actually, I have searched hard, and I cannot find the point in time that physicists shifted to the modern form of the Work-Energy theorem. Gustave Gaspard Coriolis proposed to name the integral of force over distance 'travail' - until then that integral did not have a standard name, but the work-energy theorem is not credited to him. It appears the modern form of the Work-Energy theorem grew organically. )

Joseph Louis Lagrange: Mechanics

My interpretation is that Lagrange was keen to treat Mechanics as extending the concepts of Statics. For Statics Lagrange had already been using theory of motion, but only for virtual displacements. My interpretation is that Lagrange wanted to merge the idea of evaluating virtual displacement with theory of Mechanics.

In modern physics textbooks this approach is often presented as applying d'Alembert's principle.

However, when you go to the original works by d'Alembert then you find that the concept that textbook authors offer as d'Alembert's principle is not there.

There is a set of two articles, by the historian of science Craig Fraser, with discussion of the Traité Dynamique by d'Alembert.

http://homes.chass.utoronto.ca/~cfraser/Dalembert.pdf
http://homes.chass.utoronto.ca/~cfraser/D'Alembert2.pdf

One interpretation is that it was Lagrange who introduced the concept that today is referred to as d'Alembert's principle, and that he attributed it to d'Alembert.On the history of concept of action

In my opinion the most interesting information is that Joseph Louis Lagrange did not use Calculus of Variations for problems in Mechanics.

The first to apply Calculus of Variations in Mechanics was William Rowan Hamilton, his first publication on that subject was in 1834, whereas the first edition of Mecanique Analytique was published in 1789.

In Lagrange's time another action concept did already exist: Maupertuis' action

Joseph Louis Lagrange was of the opinion that Maupertuis' action concept was not particularly relevant.

I quote from the translation of Mecanique Analytique available on Archive.org:

This principle, viewed analytically, consists of the following: in the motion of bodies which act on one another, the sum of the products of the masses with the velocities and the spaces traversed is a minimum. The author deduced from it the laws of reflection and refraction of light as well as the laws governing the percussion of bodies in two memoirs read to the Academie des Sciences of Paris in 1744 and two years later at the Academie de Berlin, respectively.

However, these applications are too restrictive to be used to establish the truth of a general principle. They have also something vague and arbitrary about them which can only make the consequence which one could draw for the accuracy of the principle itself uncertain. Thus it would be wrong, it seems to me, to put this principle as it is presented on the same level with the ones we just discussed.

Today, when people refer to Lagrangian mechanics invariably what they have in mind is Hamilton's stationary action.

However, that is not how Joseph Louis Lagrange approached mechanics. We do have that Lagrange was keen to have evaluation of virtual displacement as central concept. That evaluation of virtual displacement already gives rise to what we refer to as the Lagrangian of classical mechanics: $(E_k - E_p)$William Rowan Hamilton

David R. Wilkins has created many, many transcripts of Hamilton's articles, including the articles on classical mechanics. (The transcripts are easier to read than the scans of the originals and they are machine searcheable.)

The page titled On a general method in Dynamics is specific for giving links to Hamilton's articles on the subject of Dynamics

What today is presented as Hamilton's principle of least action is not present in Hamilton's articles in that form. I think it is safe to say that many of the thoughts that today are attributed to Hamilton were in fact not considered by Hamilton.

In his 1834 articles, when Hamilton refers to an 'Action' concept, he is referring to Maupertuis' action. Hamilton offers the name 'Characteristic Function' for his own contribution. Also, I get the impression that Hamilton was particularly interested in a formulation where the variation that is applied is variation of the start point and end point of the trial trajectory. I get the impression that that is the content of the Characteristic Function that Hamilton is referring to. Of course, the modern notion of stationary action is that the start point and end point are kept fixed, but it appears that wasn't the main interest of Hamilton.

Hamiltonian mechanics

The formulation of mechanics that is referred to as Hamiltonian mechanics relates to Lagrangian mechanics in the following way: the two are interconverted by way of Legendre transformation. One particularly relevant property of Legendre transformation is that it is its own inverse; applying Legendre transformation twice recovers the original function. This is a general property: the various formulations of classical mechanics: Newtonian, Lagrangian, Hamiltonian, are interderivable.

I recommend the 2008 article by R. K. P. Zia, Edward F. Redish, Susan R. McKay: Making sense of the Legrendre transform

While it is the case that when the Lagrangian is $(E_k - E_p)$ the Hamiltion $H$ coincides with the total energy, that is rather a fluke, as explained in the above linked article. Physics textbooks

It would appear that when an author of a physics textbook writes a "historical introduction" the author recounts from memory things they read decades before in the textbook that they learned from, written by an author who also recounted from memory, etc. However, human memory is highly error prone in situations like that. Some elements are lost, other elements are added, and the story is reshaped to a form that aligns with the author's expectation.

It is important to be aware that in physics textbooks the "historical introduction" is a highly fictionalized story. That is what happens when people do not check their memory against the original sources.

 

[Cleonis]

I have searched hard, and I cannot find the point in time that physicists shifted to the modern form of the Work-Energy theorem.

I'd like to elaborate on the above.

The development of a body of knowledge is a process of cross-pollination of ideas. The body of knowledge of Lagrangian mechanics has been developed by a community, but it ends up being all attributed to Joseph Louis Lagrange. The psychological tendency to attribute to an individual is very strong.

Remarkably, the Work-Energy theorem did not end up being attributed to an individual.

In fact, I think that not being attributed to an individual had another effect: there is no standardisation of the form of deriving the Work-Energy theorem. Some textbook authors do not mention the Work-Energy theorem at all, the ones that do give different derivations of it.

I have a strong preference for the following way of deriving the Work-Energy theorem:

We take as starting point is $F=ma$ and we integrate both the left hand side and the right hand side with respect to position, from starting point $s_0$ to point $s$
$$ \int_{s_0}^s F \ ds = \int_{s_0}^s ma \ ds \qquad (1) $$
In the next steps the following relations will be used:
$$ v = \frac{ds}{dt} \qquad \Leftrightarrow \qquad ds = v \ dt \qquad (2)$$
$$ a = \frac{dv}{dt} \qquad \Leftrightarrow \qquad dv = a \ dt \qquad (3)$$
For the time being I will not write the factor $m$, it is a multiplicative factor that is just carried from step to step. In the final expression I will include the factor $m$ again.

The integral from a starting point s_0 to final point s
$$ \int_{s_0}^s a \ ds \qquad (4) $$
At this point the acceleration profile is unspecified, but since acceleration and position are not independent we can develop (4) nonetheless.

Change of the differential according to (2), with corresponding change of limits:
$$ \int_{t_0}^t a \ v \ dt \qquad (5) $$
Change the order:
$$ \int_{t_0}^t v \ a \ dt \qquad (6) $$
Change of the differential according to (3), with corresponding change of limits:
$$ \int_{v_0}^v v \ dv \qquad (7) $$
So we have:
$$ \int_{s_0}^s a \ ds = \tfrac{1}{2} v^2 - \tfrac{1}{2} v_0^2 \qquad (8) $$
After including the factor $m$ again we use the right hand side of (8) for the right hand side of (1), thus arriving at the Work-Energy theorem:
$$ \int_{s_0}^s F \ ds = \tfrac{1}{2} mv^2 - \tfrac{1}{2} mv_0^2 \qquad (9) $$

Given the Work-Energy theorem the natural progression is to formulate a concept of mechanical potential energy: the negative of work done.
$$ \Delta E_p = -\int_{s_0}^s F \ ds \qquad (10) $$
From this definition of mechanical potential energy it follows:
$$ \Delta E_k = -\Delta E_p \qquad (11) $$
Hence:
$$ \Delta E_k + \Delta E_p = 0 \qquad (12) $$
So: the sum of mechanical potential energy and kinetic energy is conserved

As we know, over time the scope of the concept of potential energy was expanded again and again, in the sense that as new areas of physics were developed corresponding forms of potential and kinetic energy were recognized/acknowledged.

(8) is a very general result.
Here it is notated in terms of position, velocity and acceleration; it generalizes to any context where there is interplay of a value, its first derivative, and its second derivative. Also, it is not limited to derivative with respect to time.(8) expresses a relation between taking a second derivative, and squaring. I assume this relation was recognized very early in the development of differential calculus and integration. I assume it is no coincidence that the Leibniz notation for taking a second derivative uses squaring signs.

I like to think of the Work-Energy theorem as a particularly powerful synergy of $F=ma$ and (8)

Historical speculation:
The above derivation was well within the mathematical capabilities of Euler, Lagrange, and others working around the same time, yet they didn't have it in this form. How come?
I'm guessing that as soon as Lagrange had a way of relating potential and kinetic he did not look further. When you are trying to make progress and you find a tool that enables moving forward you stop looking.

 

[jonjacson]

Enter here:

https://www.17centurymaths.com/

And look for this:

A translation of Euler's Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes……… is now complete, i.e. the Foundations of the Calculus of Variations, and includes E296 & E297, which explain rather fully the changed view adopted by Euler. You can access it by clicking: Link toMaxMin.

You will see that Euler started the calculus of variations using geometry, just changing one single point in a curve. It was Lagrange who introduced the new "differential" wich Euler considered to be a step forward, so he forgot about geometry and adopted the same approach.

The history of all this is treated in Goldstine:

"A History of the Calculus of Variations from the 17th through the 19th Century"

And you can find a real masterpice if you search for this:

"Jacobi's Lectures on Dynamics: Delivered at the University of Konigsberg in the Winter Semester 1842-1843 and According to the Notes Prepared by C. W. ... (Texts and Readings in Mathematics Book 51)"

I forgot to mention this one:

https://archive.org/details/histroyofthecalc033379mbp

 

[Cleonis]

And look for this:

Back when I started researching the origins of application of Calculus of Variations in physics I soon noticed the translations of Euler works by Ian Bruce. I was elated to see the original sources made available in this way.

With Leonhard Euler mentioned, let me proceed to give information from the story: Euler and Calculus of Variations.Euler's initial foray into the realm of Calculus of Variations treated a specific class of cases: the class of cases where the function to be integrated has terms with the value itself, and terms with the first derivative of the value, and no higher order terms.

Example: the catenary problem. The two factors that go into the integration are height of the curve, as function of the horizontal coordinate, and the slope of the curve, as function of the horizontal coordinate.

There is an article (author: Preetum Nakkiran) titled: Geometric derivation of the Euler-Lagrange equation. (The derivation that Gelfand and Fomin give in their book Calculus of Variations is also along geometric lines; Nakkiran's derivation is his own.)

Here is what is interesting about that from a physics perspective:

We have for example $F=ma$, from that we derive the work-energy theorem, which gives us the definitions for mechanical potential energy and mechanical kinetic energy respectively. That is, the result of deriving the work-energy theorem gives us a way of expressing the mechanics taking place in a way that that we go up only to the first derivative (velocity) in our expression, and we no longer go up to the second derivative (acceleration).

Thanks to the work-energy theorem the Lagrangian $(E_k - E_p)$ falls within the scope of the Euler-Lagrange equation.Now:
Euler is a mathematician, and mathematicians always want to generalize.
So: Euler wanted to move to a higher order class of cases: add a term with a higher order derivative. Then generalize to encompass arbitrarily high order derivative:
$$ I[f] = \int_{x_0}^{x_1} \mathcal{L}(x, f, f', f'', \dots, f^{(k)})~\mathrm{d}x $$
(I copy/pasted the above LaTeX from the Wikipedia article)

Euler was keen to extend development of Calculus of Variations to higher generalization, but his initial geometric approach was not suitable for that.

Some time later Lagrange started sending letters to Euler, with among other things a demonstration of analytic approach to Calculus of Variations. Euler recognized that Lagrange's analytic approach accomodated the higher order generalizations naturally. That is why Euler switched from the initial geometric approach to Lagrange's analytic approach.

For the higher order cases correspondingly higher order versions of the Euler-Lagrange equation are derived.The thing is:
For physics the higher order generalizations are overkill. For what is encountered in physics the lowest order Euler-Lagrange equation is already sufficient to cover all cases.Derivation of the Euler-Lagrange equation in textbooks

We have that the physics community has converged onto a standardized way of deriving the Euler-Lagrange equation; every physics textbook with a treatment of Lagrangian mechanics gives that derivation, the one where towards the end integration by parts is applied. That derivation gets the job done, but the whole process is opaque.

For comparison, check out the geometric derivation of the Euler-Lagrange equation by Preetum Nakkiran that I linked to earlier. That derivation is informative; it tells you what is going on.

Preetum Nakkiran points out a crucial property of the Euler-Lagrange equation: it is a differential equation. A differential equation evaluates a local criterion. Therefore it must be possible to derive the Euler-Lagrange equation using only local reasoning. In his derivation Preetum Nakkiran uses differential relations only.

 

[Cleonis]

Link to MaxMin

It appears there is an error in the database that powers Ian Bruce's website, resulting in a shift.

This is the link to:
Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu acceptiThe URL of the main page, which lists all the translated works:
https://www.17centurymaths.com/

On that main page: somehow everything is shifted by one. Clicking the link that onscreen says 'MaxMin' takes you to the work that is described one entry above it. To open the actual page for the MaxMin work I had to click the link in the entry below the actual entry for the MaxMin work.

I have sent an email to Ian Bruce, describing the error.