About the Original Maupertuis Action for light

[DanielMB]

Fermat found that light in its propagation, reflection and refraction follows the trajectory with the least time, that could be expresed as a proto-action S* (not defined by Fermat)

   S* := Integral[(1/v).dl] 

   S* = (1/c).Integral[(c/v).dl = (1/c).Integral[n.dl] = (1/c).Lo

Where Lo is the optical path, so minimizing S* is equivalent to minimize Lo, besides …

   S* = Integral[(1/v).(dl/dt).dt] = Integral[dt]

So minimizing S* is equivalent to minimize the time used in the trajectory. On the other hand, originally,  Maupertuis indentified the action for light as

   S = Integral[v.dl]

How it can be reconciled?, being S* # S, is this an error in Maupertuis notes?

Please, be benevolent, I’m not a physicist. Thanks a lot
 

[quiet]

Hello DanielMB. Have you written the following?

[math] S^* = \int \dfrac{1}{v} \ dl [/math]

[math] S^* = \dfrac{1}{C} \int \dfrac{C}{v} \ dl =  \dfrac{1}{C} \ \int n\ dl = \dfrac{1}{C} \ L_o [/math]

[math] S^* = \int \dfrac{1}{v} \ \dfrac{dl}{dt} \ dt = \int dt [/math]

[math] S = \int v \ dl [/math]

[DanielMB]

Yes, that was the idea ... thanks

[DanielMB]

22 hours ago, DanielMB said:

Yes, that was the idea ... thanks

Could anybody tell me how Maupertuis based on his definition of action (S) supposedly  arrived to similar results as Fermat did it with (S*)?

 

[studiot]

2 hours ago, DanielMB said:

Could anybody tell me how Maupertuis based on his definition of action (S) supposedly  arrived to similar results as Fermat did it with (S*)?

 

I thought you were satisfied with the previous posts.

Anyway the story runs like this.

Neither Fermat,  nor Maupertius expressed their principles as integral equations.

Light was known to travel at a finite speed and Fermat thought this to be constant.

Fermat proposed that light follows the path that takes the least time to traverse.

Maupertius was dissatisfied that time took precedence over space (distance) so he proposed that light follows the shortest path.

Not suprisingly if light travels at a constant speed and takes the shortest time it's path must be the shortest path.

So they are equivalent.

 

That's really all there is to it.

[DanielMB]

On 12/9/2018 at 7:01 PM, studiot said:

I thought you were satisfied with the previous posts.

Anyway the story runs like this.

Neither Fermat,  nor Maupertius expressed their principles as integral equations.

Light was known to travel at a finite speed and Fermat thought this to be constant.

Fermat proposed that light follows the path that takes the least time to traverse.

Maupertius was dissatisfied that time took precedence over space (distance) so he proposed that light follows the shortest path.

Not suprisingly if light travels at a constant speed and takes the shortest time it's path must be the shortest path.

So they are equivalent.

 

That's really all there is to it.

No, the question was not answered yet, but yesterday while I was reading Maupertuis Memories and many letters between him and Euler about the theme,  I have arrived to the correct answer.

 I’ll use the following reference …

 1) Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles

       https://fr.wikisource.org/wiki/Accord_de_différentes_loix_de_la_nature_qui_avoient_jusqu’ici_paru_incompatibles

 

In Reference #1, Maupertuis says (in the paragraph @423)

 

… "elle prend une route qui a un avantage plus réel: le chemin qu’elle tient est celui par lequel la quantité d’action est la moindre."

 

"Il faut maintenant expliquer ce que j’entends par la quantité d’action. Lorsqu’un corps est porté d’un point à un autre, il faut pour cela une certaine action, cette action dépend de la vîtesse qu’a le corps & de l’espace qu’il parcourt, mais elle n’est ni la vîtesse ni l’espace pris séparément. La quantité d’action est d’autant plus grande que la vîtesse du corps est plus grande, & que le chemin qu’il parcourt est plus long, elle est proportionnelle à la somme des espaces multipliez chacun par la vîtesse avec laquelle le corps les parcourt."

 

 … "(The light) takes a trajectory that has a real advantage : the followed trajectory is that with the least amount of action."

 

"It is mandatory to explain what I understand by action quantity. When an object is carried from one point to other, does exist a certain action, this action depends on the velocity of the object and the length of the followed trajectory, but it is not the velocity nor the length taken separately. The action amount is bigger as the object velocity is also bigger, and when the trajectory increases its length, IT’S PROPORTIONAL TO THE SUM OF THE LENGTHS MULTIPLIED BY THE VELOCITY OF THE OBJECT"

 

The text I highlighted in uppercase means (or could be interpreted as) the integral of v.dl

 

That’s wrong!!

 

For the light, the integral that minimizes the time is (1/v).dl and not v.dl

 

The reason is the following :  In Reference #1, Maupertuis says (in the paragraph @422)

 

 “Descartes avoit avancé le premier, que la lumière se meut le plus vîte dans les milieux les plus denses”

 

“Descartes had advanced at first that LIGHT MOVES MORE QUICKLY IN MORE DENSED MEDIA”

 

Maupertuis followed Descartes & Newton in his maths, so commited the error of evaluating the optical path with that incorrect hypothesis, because LIGHT MOVES MORE SLOWLY IN MORE DENSED MEDIA. That's the reason because I could not understand Maupertuis PLA in his enunciation

Thanks

 

[studiot]

Thank you for the reply, but I don't quite read the same line of reasoning into your extracts.

In fact I am not quite sure what you are saying in several places.

The 'action' Maupertius refers to is quite different from our use of the word today as the difference between kinetic and potential energies.
Indeed the word 'energy' had yet to be formalised or even introduced.

But Snell's law was the 'acid test' that distinguished between Newton's theory that of Young.

[DanielMB]

@Studiot

I know Maupertuis' action is different from the modern concept of action, in fact when the total energy E is conserved, the Hamilton-Jacobi equation can be solved, emerging Maupertuis' abreviated action

What I've questioned is the first Maupertuis definition of action as the sum of <<<< v.dl >>>>>>, this definition does not minimize the time insumed by light in the trajectory, the correct abreviated action for the light should have been defined as  <<< (1/v).dl >>>> ,  this definition does minimize the time insumed by light in the trajectory!!!!!

And the cause of Maupertuis' error was to believe (following Descartes) that light moves quickly in a dense media than it does in vacuum

Notes : Hero of Alexandria arrived to similar conclusions as Fermat did, but Hero only analyzed propagation and reflexion, not refraction, while Fermat included refraction, Maupertuis created and added epistemological significance to the term "action" an with Euler extended the scope of the principle

 

Edited September 14, 2018 by DanielMB

[studiot]

You have quoted short extracts from what was presumably Maupertius' original writing.
These do not describe the intended circumstances of transmission of the light.
Perhaps a fuller account would do so (in French and/or English)?

I only have access to latter day English descriptions which suggest that the intended circumstance was passage through only one medium at a time and that refraction on passing from one medium to another only resulted in a change of constant speed. That is Maupertius did not deduce Snell's Law.

If that was the case then my comment about least time being equivalent to least distance still stands.

It was also suggested that Maupertius changed (elaborated on) his proposal after Fermat and following discussion with Euler.

So dates of statements might be of importance.

 

Fermat's Principle itself was originally stated as a minimum condition, but we also now know that this should be an extremal condition since it is possible to devise light paths that are a maximum. For example some reflections at a concave elliptical mirror.

Edited September 14, 2018 by studiot

[DanielMB]

@Studiot

Thanks for your answer.

Maupertuis deduced Snell's laws (reflexion and refraction laws), you can see it in this link :

   https://fr.wikisource.org/wiki/Accord_de_différentes_loix_de_la_nature_qui_avoient_jusqu’ici_paru_incompatibles

Search for the paragraph #424, explicitly Refraction Law but minimizing and incorrect funcion : v.dl, so he arrives to the law , but using as refraction index n = v/c (incorrect) and not n = c/v (correct), just because his respect to Descartes and Newton (v>c in optical densed media -obviously incorrect-), so we had to wait until XIX century for the Fizeau experiments to elucidate that the corpuscular theory of Newton & Descartes was wrong, and light moves slowly in a densed media, but not Fermat nor Maupertuis, even Euler knowed that ...

Yes, you are right when you express that the "minimization" could be in a certain specific scenarios valid as "maximization", and others where the "critical point" is a saddle point, as it is showed in this Gray's paper:

   http://www.eftaylor.com/pub/Gray&TaylorAJP.pdf

So, being rigorous, LAP (Least Action Principle) should be abreviated as CAP (Critical Action Principle)

It's rather difficult to read Maupertuis in the web, because most of the letters to the french Academy are in french, Euler sent his letters to Maupertuis in latin and Maupertuis answered then also in french ....

Edited September 15, 2018 by DanielMB

[studiot]

Thanks it will take some time to read those links, but I will come back to you.