Fermat's Principle of least time: Reflection by Fermat's principle Photo Credit: mdashf.org

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Fermat’s Principle, a lecture in optics

Optics series lecture, Lecture-III

“Geometrical Optics and Fermat’s Principle”.

Geometric Optics: When the size of objects that a wave of light interacts with are large compared to the wavelength of light λ, λ can be neglected for practical purposes and the light waves behave like rays of light. Rays of light are geometric line segments from one point of incidence of light to another. Study of optics under the limit of negligible wavelength λ → 0, is called geometric optics.

Geometric optics can be studied using Fermat’s principle, much like motion of objects in the realm of classical mechanics are studied using Newton’s laws of motion. To know the basic grounding of Fermat’s principle follow the links to read two articles which expound the subject matter of Fermat’s principle, article 1 — a detailed, historical and kind of long article, and article 2 — a conceptual but a short article.

Before Fermat, Hero of Alexandria, who lived sometime between 150 BC and 250 AD explained reflection of light. ( Read the more extensive history in the already linked article, article 1 above ) His formulation is stated as principle of shortest path.

Since reflection occurs in only one medium ( homogeneous medium ) light indeed travels a geometric shortest path; this is the straight line path between any two points — or coordinate of the ray. For homogeneous medium optical path and physical and geometrical path are merely either proportional to each other or equal.

In the modern times Fermat reformulated Hero’s principle of shortest path — to its equivalent form of shortest optical path. This entailed the principle to be applicable to both reflection and refraction and any other possible optical phenomena which could be explained by virtue of Fermat’s principle in general.

In its original — shortest path form the principle could not explain refraction, because the latter involves traversal of light rays in in-homogeneous media, that is different media are traversed at different speeds and optical path and geometric or physical path are no more equivalents.  We will soon see this in detail.

The new formulation of Fermat which is based on improvement of the earlier Hero’s principle for reflection is called as Fermat’s principle of least time. It states that “a ray of light travels through those coordinates of the ray in a given system of media of varying refractive indices for which the amount of time taken is least .”

This can successfully explain both reflection and refraction. But it can still be generalized and the modern form is in terms of the shortest optical path which is different from how it was originally formulated. Before we study the modern form lets discuss its original form.

According to Fermat “the ray of light will correspond to that path for which time taken is an extremum in comparison to nearby paths” Mathematically extremum implies time for a particular path can be minimum, maximum or stationary for a given neighborhood of paths.

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Inherent ability = difficulty * accomplishment.

All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.

First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.

fixed: which changes only if estimated wrong.

That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics.

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Blackholes and the Action Principle.

This is not an article that purports to establish any link between what I have to say about Blackholes and what I have to say about Action Principle.

First off there are always a connection between action principle and anything else in Physics. Everything in principle can be derived from the action principle. The action principle is the general principle of all sorts of mechanics in Physics and has the weird power of superiority over everything else, at-least in principle. So if you want to claim something is inconsistent in Physics, take a piece of chalk and chalk out how it is that the action principle is inconsistent with anything that you want to discard. The action principle has the key to proving how consistent or inconsistent Physics is.

And I am not stating the action principle much, either. I just wish to mention two ideas that have no explicit connection to be brought out by this article. Hence they are to be read without any purported conjunctions of one with the other.

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Mirror and Fermat's principle: We can see ourselves in the mirror and take our mirror reflected selfie as a consequence of Fermat's Principle, the topic of discussion of the blog. Photo-credit: weddingwire dot com

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Optical Path and Fermat’s Principle.

Mirror and Fermat’s principle: We can see ourselves in the mirror and take our mirror reflected selfie as a consequence of Fermat’s Principle, the topic of discussion of the blog.

Snell’s Law governs refraction which is adjustment of optical paths in in-homogeneous media because light can no more travel at its speed in free-space.

Snell’s Law and Refraction.
The above expression comes from Fermat’s optical theorem, called as “Fermat’s least time principle” which can in turn follow either from;

a. Huygens’s Principle; that light travels like spherical wave-fronts hence satisfies geometric rules or,

b. Principle of least or stationary action ( or Hamilton’s action principle ).

So in the beginning of our understanding we thought “Light travels a path which is shortest, or the least-path”. This is due to Heron of Alexandria, who lived between 10 AD and 70 AD. Then this path which is traversed by light was redefined to something called optical path, which led us to our understanding that “light travels the path of least time”.

All physical laws are derivable mathematically –with appropriate physical understanding, from Hamilton’s Action variation or Principle of Action, made to provide the least or stationary time. Then time is replaced by definition of action as the most general formulation of the law.

So all in all, path/distance >> Time >> Action . That is crudeness goes towards abstract physical understanding. This can be recognized as an attribute of unification and tells us why Mathematics reigns supreme in Physics.

In the last few weeks I am trying to understand why light traverses straight lines and why it refracts. The other day, I saw a little mug, floating inside a bucket full of water. Inside water any object would look shortened, this is known by a phenomenon called as refraction.

Lets envisage the phenomenon or observation, via this quickly reproducible trick. Lets dip our favorite pencil in a glass of water and another in a glass filled with air. This is how it looks, I quickly made this arrangement to which my 3 year old niece just glued with rapt attention.

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Why is energy conserved?

Here is the way; that hasn’t been changed in a long time. We, start with a simple object, and we note that, such an object is defined for its motion by whats called, its location in space = x, or, the increments in its location, which is called, an infinitesimal distance = dx, the instantaneous time at which its motion is referred, t, or, the increments in its time, called dt.

Thats it. And, we would like to know; all that the object does in terms of x, t.

I would not like it, if my dear people sit in a car, and it vanished into thin air and never came back. I would like to keep track of it, the car, because I know my people would still be in it. I would like to, keep track of satellites, and, missiles and airplanes, I would like to know, whats happening around me, and why its happening.

It all started, with the quest, to solve for the trajectory, and then, became more complicated, as the complexity of these objects or systems grew. Collectively, they satisfy greatly, the quest we had set on, since the millennium and more, to understand, whats all, that goes on, in our universe, in our close vicinity, and in situations far off from us, as far as the extraneous bounds of the galaxy, in which we live, and more and more and more and deeper and deeper.

Then, x, t are not sufficient, to describe such situations. But, since its all systematic, we know all that has been defined, its not a party or Ramstein Music Band, where you forgot what happened yesterday. Its Hello Physics Inc. Pay Attention.

Now, as we defined dx and dt, we also note that, their ratio, or as-is-called, rate of x wrt t, called speed = v, is often formally written, as, x with a dot on it, … Its the first order time derivative of x = …

We also form, two quantities;

1. from v we form m.v = p = momenta, by multiplying the mass m into the velocity or speed v.

2. from a, we form in the same way, F = m.a = force.

But, force is also defined to be the time-rate of change of momentum p, or in other words, the ratio of the increments dp and dt, …

This latter is called Newton’s 2nd law; …

Its called a law, but strictly speaking, its a mathematical law so far, and not one which describes, universe’s phenomena so it cannot be called, a physical law or principle as of yet.

This point was originally raised by Feynman; as far as I know.

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