The principle of stationary action from the previous post on Lagrangian mechanics can also be used outside pure mechanics. For example, for traveling light, including when it undergoes reflection or refraction. In this case, the action is not an energy but rather the time taken to traverse the distance under consideration. The principle states that the path taken will be the one that reduces the time taken to the least possible out of all possible paths, while respecting the physcial constrains of the system.
Example 1: Point light source
The time taken will be least if light travels in the shortest path between two possible points (if the velocity of light is constant throughout the path) so it implies that the travel path will be straight lines. If it takes any other path, it has not reached the end point (or any of the points enroute) in the shortest possible time. So a point light source emits straight rays in all directions outwards. However, the light remains straight only if the speed of light in the medium does not change! For example, in the Feynman lectures it is pointed out that when light traveling from a source to a person encounters cold air, it will bend to pass through another layer of warmer air faster and then finally curve back to the person’s eye level to maintain the fixed destination of the path.
Example 2: Reflection on a smooth surface
Even if the surface is curved, it will locally be flat at the point where a ray strikes (since it is smooth), so the curvature makes no difference to the law of reflection itself (only to the image when a bunch of rays strike different points on the surface). The angle at which the ray strikes the normal to the point of contact on the surface is called the incidence angle.
Firstly note, after reflection, the light has to continue in a straight line for reasons similar to the previous example. Let’s call the angle of this reflected ray the angle of reflection, t. Let c be the speed of light in the medium and d be the normal distance between the tangent to the surface and a point source. Then the time to be minimized is the sum of travel times before and after reflection and the constraint is that the distance covered has to be constant. Since the principle holds for any point in the reflected ray, we can easily pick the point where the vertical distance covered is zero and the horizontal distance covered (along the tangent to surface) is the constraint.
T= d/c (sec(i) +sec(r)) ……(1)
tan(i)+tan(r) = constant/d = constant …(2)
Differentiating (2), we get
dr/di= -sec2(i)/sec2(r) … (3)
To find the extremum of (1), we differentiate w.r.t to “i” and set to zero,
sec(i)tan(i)+sec(r)tan(r) dr/di= 0
Using (3) in above, we get that sin(i) = sin(r), or the law of reflection that the angles are equal!
Example 3: Refraction between non-dispersive media
Since the light has different velocities c1 and c2 in the two media, at the surface separating the media, the light refracts in order to maintain the shortest time principle. The governing equations would be as below and the minimization method would be similar to example 2. Note that again, without loss of generality, we have chosen a point at a far enough normal distance in the second medium that the refractive surface is equidistant to the chosen point and the point light source in the first medium.
T= t1+t2 = d/c1*sec(i) + d/c2*sec(r)
c1*tan(i)+ c2*tan(r) = constant (x/d)
Snell’s law: sin(i)/sin(r) = c1/c2 would yield the desired solution.
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