# Could someone explain to me Fermat's theorem?

I DO NOT like this least time hocus pocus. I prefer the idea of causality. I just CANNOT stomach this idea. Here are my arguments (italicized text) almost verbatim from my notes against what I read (in bold). Someone please explain to me the whys and hows.

Arguments against Fermat:

"Given a source point A and receiving point B, light takes path of least time to get there"

<-----A B

If the light ray is sent as in my case, it will never reach B. In such a case a path of least time does not even exist. But if I simply took Snell's law/ angle of incidence = angle of reflection law, I would be able to say point by point where it is going to go. I find that simpler to think of.

Correction/ better wording of Fermat's theorem that was made a little later:
"light takes such a path such that the paths infinitely close to it take nearly the same time"

Light takes such a path? A path to go where? How do you know WHERE the light is going? How do you know it's final destination? Using causality, there was no confusion. At every instant I could calculate WHERE the light was heading. Seems to me you have already pre-determined where it ought to go. I don't get that at all.

Why the heck would it want to go to a point B? What's so special about it? Just because you are sitting there, or you WANT light rays to reach there doesn't make it any more special. Only if some light is HEADING your way will it come to you.

I can do sums in the Calculus of variations, if you need that to explain to me what's going on. I did the chapter in Boas before I started this stuff.

If you don't understand what I've written above and think that I am not clear with what I am trying to say I don't understand, here is an extract my Feynman lectures, following which I have presented the same argument in different words.

Feynman is explaining the mirage caused in deserts and hot roads. [Chapter 26, Volume I]

....Why[is the mirage caused]? The air is very hot above the road but it is cooler up higher. Hotter air is more expanded than cooler air and is thinner and this decreases the speed of light less. That is to say, light goes faster in the hot region than in the cool region. Therefore, instead of light coming down in the straight forward way, it also has a least-time path by where it goes faster for a while in order to save time. So, it can go in a curve...

1. First of all, he seems to be fixing yourself as a point that light rays must come to.

Why should that be so? Why should the light WANT to come somewhere?

If I had a point source, I could decide to send light only in one direction, away from you. Instead of using any least time thing, I could set up a mirror in the direction I am sending the light rays towards and thereby make them reach you. Here, the light didn't WANT to come to you. I made it do so.

2. How do you know a path of least time exists in the first place? Using what I said above, if I didn't have a mirror in the above case, it wouldn't even head in your direction. In such a case a path of least time wouldn't even exist.

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Born2bwire
Gold Member
You really can't take Snell's Law here because Fermat is more basic than Snell's, you find Snell's Law using Fermat's principle.

There is no causality with Fermat's principle, any attempt to apply it creates glaring problems. Fermat was not saying that the light actively seeks out the shortest path, he is simply saying that if you have a light source at point A and observe the light and point B, then the paths that the light takes from A to B follow the shortest time. You cannot say that given light source at A, then the light follows the shortest time path to get to B. Your attempts to apply causality to the theorem is why you are arriving at these questions.

For example, the answers to your last two questions are simple. If there is no light observed then there is no path. We need light to be observed at a point for any path to actually exist. So if we observe light, then we know that a path exists and we apply the least time principle to deduce the paths taken by the light rays. If no light is observed, then we know that no possible path exists. On a similar note since you mention Feynman, Feynman's path integrals are somewhat analogous to Fermat's principle and Lagrangian mechanics. In both we are looking to minimize a value or metric in order to find the path of motion (technically we are finding the stationary path, but a minimum path is often a subset of the possible stationary paths, see edit). In fact, Feynman's path integral formulation for quantum electrodynamics will become the same as Fermat's principle in the classical limit (and the classical Lagrangian paths of motion in the classical limit for particle motion).

EDIT: I should also mention that the more modernized statement of Fermat's principle states that light follows the stationary path. There are a few cases where the shortest time path is not the correct path, but if you were to find the stationary path then it is correct. This quality is reflected in Feynman's path integral formulation.

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You're confused by the anthropomorphism that you are introducing yorself. Just because the answer turns out to be the path of least time, that doesn't mean that the light "wants" to travel in the least time.

You really can't take Snell's Law here because Fermat is more basic than Snell's, you find Snell's Law using Fermat's principle.

There is no causality with Fermat's principle, any attempt to apply it creates glaring problems. Fermat was not saying that the light actively seeks out the shortest path, he is simply saying that if you have a light source at point A and observe the light and point B, then the paths that the light takes from A to B follow the shortest time. You cannot say that given light source at A, then the light follows the shortest time path to get to B. Your attempts to apply causality to the theorem is why you are arriving at these questions.
Oh I see. But if I did shoot out light from point A, and wanted to find out what path it took, then I cannot use this method? Essentially, when and how do I apply this method?

EDIT: I should also mention that the more modernized statement of Fermat's principle states that light follows the stationary path. There are a few cases where the shortest time path is not the correct path, but if you were to find the stationary path then it is correct. This quality is reflected in Feynman's path integral formulation.
Yes. I have read in certain places that this is very similar to Lagrangian mechanics. Feynman himself mentions it somewhere in that lecture. I did mention the correction that the path must be stationary in my post.

However, what I don't see is why I can't apply Snell's law if it is a specialized subset of Fermat's.

Born2bwire
Gold Member
You can use this method certainly, but I think you just need to be aware of the implications of what the theory implies. The theory is not meant to be the cause for observing light at a point, just the result. You can most certainly apply the theory without knowing if you can observe the light at point B, but I guess you could not say, a priori, that the path you found is correct until you could prove that light would be observed.

For example, let's say that the source is placed at point A and the observer is at B. In between A and B in direct line of sight is an impenetrable wall. But, the wall is not infinite. So, the Fermat's theorem would dictate a path that would go around the wall, maybe so far as to actually traverse the perimeter of the wall until it gets to the other side and then it goes straight to B. Either way, you will find a path, however, common sense will tell you that because of the wall blocking your line of sight with A, no light will be observed at B. So while a path is found, it is not a valid one because observation is in fact impossible. However, the circumstances in which no path will ever exist are easily deduced without actually having to do an actual experiment. Like knowing that light will not change direction unless it meets an inhomogeneity (which is why the light can't go around our barrier).

So the point is, you can always apply this method, but you it will only work if light can actually be observed. The circumstances where it will not work are easily seen by our own experience. Take a look at the derivation of Snell's Law to see a way to apply Fermat's principle. It is like an optimization problem. You have an environment and given a light source, you just need to figure out the paths of least time to find how the light will propagate.

You can use Snell's Law and the rules of specular reflection, but what I meant was that since they are found by using Fermat's principle, using them is no different from using Fermat. So if you have problems with the validity of Fermat's principle, these problems extend also to Snell's Law.

Staff Emeritus
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There is no causality with Fermat's principle, any attempt to apply it creates glaring problems.
Exactly. How does a thermos know to keep hot liquids hot and cold liquids cold?

I see. So this principle does not tell you WHERE anything goes. It is just that IF it happens to go there, which you will have to deduce some other way, then in that case you know the path. It doesn't seem very useful to me. The causal way of looking at "stuff" seems better. Why is it so important in mechanics?

jambaugh
Gold Member
I understand the reticence to accept action principles like Fermat's principle. It seemed like Voodoo to me to at first. But the issue is that one is trying to reconstruct the (causal) dynamics, not explain it. Think of Fermat's principle as encoding in a very concise fashion the dynamics of the propagation of electromagnetic waves. When someone says "we derive Snell's law" from "Fermat's principle" they are (should be) saying we decode the practical application of the dynamics encoded within FP.

You can go back and forth proving Fermat's principle from Huygen's principle and vis versa. Huygen's principle is more causal in the sense you seem to be seeking. But its messy to apply if you just want to get the right geometric optics answers.

If it helps remember (in the generalization of Lagrangian mechanics) one is really seeking to pick the action functional which when minimized recovers the desired causal dynamics. It is an efficient point at which we make guesses about how nature behaves because it must manifest the symmetries (and corresponding conservation laws) we see in the causal behavior.

It seemed like Voodoo to me to at first.
HURRAY! I'm not alone. My sentiments exactly.

But the issue is that one is trying to reconstruct the (causal) dynamics, not explain it. Think of Fermat's principle as encoding in a very concise fashion the dynamics of the propagation of electromagnetic waves. When someone says "we derive Snell's law" from "Fermat's principle" they are (should be) saying we decode the practical application of the dynamics encoded within FP.

You can go back and forth proving Fermat's principle from Huygen's principle and vis versa. Huygen's principle is more causal in the sense you seem to be seeking. But its messy to apply if you just want to get the right geometric optics answers.

If it helps remember (in the generalization of Lagrangian mechanics) one is really seeking to pick the action functional which when minimized recovers the desired causal dynamics. It is an efficient point at which we make guesses about how nature behaves because it must manifest the symmetries (and corresponding conservation laws) we see in the causal behavior.
Man! I am having a hard time with this stuff. Could you please explain a little more detailed?

Born2bwire
Gold Member
Man! I am having a hard time with this stuff. Could you please explain a little more detailed?
I think it just comes down to how you use it as a tool. Fermat's Principle will work, just like Lagrangian mechanics, but the process is just not to be taken as a straight-forward physical one. I guess it is the difference between saying, "I place a mass M a distance R from the Sun and give it an initial velocity v, the mass will move in accordance to Newtonian gravity and it will, using the relationship between forces and after much calculation, follow this path." as opposed to saying, "I place a mass M a distance R from the Sun with an initial velocity v, the mass will move along the stationary path given by Lagrangian mechanics." The difference here is that we went with the direct forces to derive the path of motion in the first part and in the second part took the model of the system to find the path of motion. The end result is the same but the physics of WHY the mass moved the way it did is explained by Newtonian gravitation. Both methods use gravity (in the Lagrangian case it would be hidden in the potential energy) but the first method gives us a direct relationship when we work out the problem.

Either way, they both give you the same result but in many problems, the Lagrangian mechanics formulation is much easier. Just like the path integral is much easier in certain quantum mechanic problems and Fermat's Principle helped us distill Snell's law theoretically in a manner that is much easier than using Huygen's principle. Like jambaugh said, you could probably use Huygen's principle to solve the same types of problems that you would use Fermat's principle. It would give you a more realisitic picture of the physics but in terms of solving the problem it will probably be much more difficult.

Here's what I've understood about the Lagrangian / Fermat's principle :

1) So it's just a mathematical trick to get the path of the motion given the system.
2) It does not give any more intuition (at least not to me!) about the mechanics involved.
3) In Lagrangian mechanics/ Fermat's you need to assume a final destination point to calculate that stationary path. Likewise, like you mentioned above, IF I see the light at point B, I can tell which path it came by. Therefore it is more a-posteriori in comparison to the Newtonian method.
4) Having got the "description" of that path, you can vary your end point and adjust the mathematics/formulae as needed.

How's that? Have I understood what you've said?

Could someone please confirm if my above summary is ok?

Born2bwire
Gold Member
Yeah a lot of it I would agree with. Though I would reiterate that the principles are still grounded in physics. There are many problems where you still need to input some sort of external reasoning into solving them like you do here. For example, let's say I have a valley between two hills and I place a boulder at some height less than the peaks between the hills. What is the speed of the boulder 2 m lower when it reaches the side of a hill that does not face the valley? Well, a naive answer would be that you find the change in potential energy associated with 2 m and convert it into kinetic energy and solve for the velocity. The correct answer is that the boulder can never travel to a side of either hill that does not face the valley since it does not have enough total energy to overcome the crests. This is the kind of thinking that you have to inject into the same kind of problems here with Fermat's principle.

As for the third and fourth point, you could combine the two. That is, you can directly solve for an arbitrary point x. The solved path will require certain constraints on x that restrict it to the stationary path. For example, in Snell's Law, you solve for a two-dimensional path and in the end we define the refracted angle in terms of the incident angle. A path of motion would normally be defined as a function of time and so on. So you can solve for the correct arbitrary path, you are not restricted in saying I have the source at A and observe at B solve for the path. Ok, now I move the observation to point B' and solve. Ok now I move... You do not need to do this as an explicit procedure, it can be solved in one go.

Great. I am still not too happy with it, but getting better.

I always thought that "least" (or stationary) action type concepts or principles were kind of intuitively friendly when we consider that generally, we usually expect things to "move" or progress from one state to another with a natural economy. It makes sense that a physical system would tend to take the least (or most efficient) path through a space of its possible configurations. I guess it's a matter of personal perspective. I'm kind of lazy and I expect nature to be like me :-)

jambaugh
Gold Member
Man! I am having a hard time with this stuff. Could you please explain a little more detailed?
Sure,
Consider the various ways we can express the same information:
"The set of points one unit away from the origin in the x-y plane."
"The set of ordered pairs (cos(t),sin(t)) where t ranges from 0 to 2pi"
"The set of solutions (x,y) to the equation x^2 + y^2 = 1"
"The set of complex numbers you get by raising i^x for any real x."
...
All express (encode) the same information but do so differently. Mathematically we can derive each from the other, they are equivalent but their meanings in various contexts may reflect the application. If you are trying to draw the circle you may find the parametric form more natural. If you are thinking in terms of groups then the complex power form may be more natural. One form may more naturally apply to your application while another may be more conscise. For example we can represent points in the plane very efficiently as complex number but in the application we really are using them as vectors.
(I like deriving the centrifugal and coriolis acceleration terms by differentiating [itex]z = re^{i\theta}[/tex]. I then convert back to vectors.)

Hugen's principle states that the dynamic evolution of a wave can be equated to treating each point in the wave front as a fresh distrubance (source). It is a nice starting point if you are dealing with diffraction. It also reflects the local physical interaction of the propagating medium or field. It is in this sence more "natural".
However it is a messy starting point for doing geometric optics and refraction.

Dual to a propagating wave fronts or more precisely dual to the lines (or in 3-d surfaces) of nodes and antinodes you usually visualize when thinking of a wave, are the set of normal curves which run in the direction of the wave propagation. For example if you have spherical waves propagating from a point source you will have these dual radial lines radiating outward from the point source.

If you know the radiating lines you can reconstruct the wave-fronts and vis versa. With a heavy bit of math you can derive conditions on these lines equivalent to the Huygen's principle for the wave fronts. In particular since each point on the wave-front is acting as a source you get that the lines are the shortest (time-wise) paths the propagating wave takes. It is a matter of constructive interference for the many paths taking by all the sources along the wave-front. Thus Huygen's principle can be re-encoded as Fermat's princple. Also you can go in reverse deriving Huygen's principle from Fermat's. The two are effectively equivalent ways of expressing the same dynamic constraints on the propagating waves.

If you are doing diffraction or interference problems you'll usually find it best to work with Huygen's princple but if you are doing refraction/reflection problems as in optics you generally will rather work with Fermat's principle.

It only seems like Voodoo. It is really hard mathematical physics at work.

If you know the radiating lines you can reconstruct the wave-fronts and vis versa. With a heavy bit of math you can derive conditions on these lines equivalent to the Huygen's principle for the wave fronts. In particular since each point on the wave-front is acting as a source you get that the lines are the shortest (time-wise) paths the propagating wave takes. It is a matter of constructive interference for the many paths taking by all the sources along the wave-front. Thus Huygen's principle can be re-encoded as Fermat's princple. Also you can go in reverse deriving Huygen's principle from Fermat's. The two are effectively equivalent ways of expressing the same dynamic constraints on the propagating waves.
Well, that was one of the things that did confuse me. Fermat's principle, like I stated earlier is more a-posteriori in that sense. You seem to go backwards AFTER finding out how the wave moves to develop a model on it. In the examples you gave pertaining to the circle though, I could start off with ANY one of them and they all amount to the same thing. I'm not worried about which method when used will be easier to solve problems (not right away at least), but am more worried now about understanding the physics.

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Well, that was one of the things that did confuse me. Fermat's principle, like I stated earlier is more a-posteriori in that sense. You seem to go backwards AFTER finding out how the wave moves to develop a model on it. In the examples you gave pertaining to the circle though, I could start off with ANY one of them and they all amount to the same thing. I'm not worried about which method when used will be easier to solve problems (not right away at least), but am more worried now about understanding the physics.
What does understanding the physics mean? If it means you want a physical mechanical explaination based on your personal experience or analogy (its like a ball rolling down a hill) you will run into a wall eventually on almost any subject in the sciences. I have run into the same problems you bring up on different subjects. I am very glad you brought this up though, as I had read the lifeguard problem before and realize (again) I was not applying it in the proper manner in my thinking. This really was an excellent thread, I appreciate the questions and the input a great deal.

What does understanding the physics mean? If it means you want a physical mechanical explaination based on your personal experience or analogy
I do want a mechanical explanation based on my previous experience but my experience includes Newtonian Mechanics not just a lay mans understanding. I'm not looking for any gears behind the engine. I just wanted something a little more intuitive, or more precisely, causal. I have (now) learned that this cannot be so. It is not causal. Period.

You don't consider forces acting every instant and so on, you analyze the system as a whole. You DO require quite some intuition about the subject - I realized this after reviewing the problems in Boas, and saw that I'd done each problem knowing SOMETHING about the system. Like I said, I'm still coming to grips with this method; not yet at peace.

I am very glad you brought this up though, as I had read the lifeguard problem before and realize (again) I was not applying it in the proper manner in my thinking. This really was an excellent thread, I appreciate the questions and the input a great deal.
You're welcome.

I do want a mechanical explanation based on my previous experience but my experience includes Newtonian Mechanics not just a lay mans understanding. I'm not looking for any gears behind the engine. I just wanted something a little more intuitive, or more precisely, causal. I have (now) learned that this cannot be so. It is not causal. Period.

You don't consider forces acting every instant and so on, you analyze the system as a whole. You DO require quite some intuition about the subject - I realized this after reviewing the problems in Boas, and saw that I'd done each problem knowing SOMETHING about the system. Like I said, I'm still coming to grips with this method; not yet at peace.

You're welcome.
Then start with gravity. We have some very mechanical predictions and some rules. But I have absolutely no mechanism. I cant see gravitational fields. And when I type fields or waves, I realize I am already wading into analogy that is beyond my mechanical understanding. We dont have the sensory tools. But we have math. And as long as we can use it, and it is predictive, we have achieved something.

I was not trained a Physics guy. So I rely more on a theme of what we can and cannot do as humans... animals that have gotten themselves into very deep water. I am very satisfied with the amazing predictive nature of the language of math. But remain deeply humble about what exactly we can and cannot understand through senses put to reason.

Then start with gravity. We have some very mechanical predictions and some rules. But I have absolutely no mechanism. I cant see gravitational fields. And when I type fields or waves, I realize I am already wading into analogy that is beyond my mechanical understanding. We dont have the sensory tools. But we have math. And as long as we can use it, and it is predictive, we have achieved something.
Oh. Don't worry. I got the "we just can't grasp certain things" bit. But there's a good reason why mathematicians don't make the best physicists I suppose, even with their ability to handle formulae.

Oh. Don't worry. I got the "we just can't grasp certain things" bit. But there's a good reason why mathematicians don't make the best physicists I suppose, even with their ability to handle formulae.
And vice versa. They need each other for certain types of problems. The math people sometimes point out that some outcomes cannot be thrown out just because they dont make physical sense. Then occassionally, the Physics guys find out the value or finding obtained from the math actually accounts for something yet unforseen. And then go and try to invent an experimental setup that might detect the unforseen but now reasonable finding. It is quite an exciting interplay I should think for each. Most of it is beyond me, but I get a little bit of feeling for it from some classical stuff.

Anyhow... thanks for bringing it all up in the first place because I learned something.

No problem.

And I in turn thank the guys who answered my questions, although if the discussion were to continue further from here, it would be most welcome.