This article is part of a set of three; the common factor is Calculus of Variations. In classical physics Calculus of Variations is applied in three areas: Optics, Statics, and Dynamics. Each article in the set is written as a standalone article, resulting in some degree of overlap.
The other two articles:
Statics: the Catenary
Dynamics: Energy Position Equation


Fermat's stationary time

When light transitions from one medium to another there is refraction. This refraction is described by Snell's law. Snell's law is inferred from the experimental data.

Christiaan Huygens proposed a way of understanding Snell's law in terms of reconstitution of a wavefront. It is common to refer to this idea as ‘Huygens' Principle’. I actually prefer to use the designation ‘wavefront hypothesis’.

(In my opinion the qualification ‘Principle’ is used too often. If everything is a principle then the word ‘principle’ is rendered meaningless.)

It is assumed that the wavefront is always perpendicular to the direction of propagation.

Picture 1. Image
Refraction by wavefront reconstitution

In the diagram the length of the line segment 'd' is not important; the value of that length is necessary for calculation, but in the course of that calculation the value of 'd' drops out.

In the time interval 't' the light travels a shorter distance in the denser medium, in the proportion of v2/v1. So: the wavefront hypothesis gives the following expression that is equivalent to Snell's law:

\frac{\sin\alpha_2}{\sin\alpha_1} = \frac{v_2}{v_1}
(1)

Rectangular triangles

To set things up for discussing Fermat's stationary time I must first discuss a geometric property of rectangular triangles.

Picture 2. Graphlet

We will need an expression for the rate of change of the length of line segment C as the line segment A is shortened and lengthened. So we set up differentiation of C with respect to A:

\frac{dc}{da} = \frac{d(\sqrt{a^2 + b^2})}{da} = \frac{a}{\sqrt{a^2+b^2}} = \frac{a}{c}
(2)

With the intermediate steps removed:

\frac{dC}{dA} = \frac{A}{C}
(3)

(3) is a geometric property that is not specific to optics or even physics, it is a mathematical property.

Fermat's stationary time hinges on this geometric property: the rate at which the line segment C changes in response to change of line segment A is determined exclusively by the ratio A/C.



Picture 3. Image
Fermat's stationary time

In Image 3 the letter 'S' stands for ‘Snell's point’. We will take as our starting point that there is a fixed point from where the light is transmitted, point 'T', and that there is a fixed point 'R' where the light is received. (T and R not shown in the image; T and R can be arbitrarily far away.)

If it is granted that the wavefront is perpendicular to the direction of propagation it follows that the angle β1 is equal to the angle α1, and that the angle β2 is equal to the angle α2.

The variation of the path of the light consists of moving point S along the refraction line. We want to find the criterium that identifies the location of point S in the variation space such that Snell's law is satisfied.

Moving point S changes the A1/C1 ratio and the A2/C2 ratio.

We have established earlier that the A/C ratio, which is the sine of the angle, is equal to this derivative: dC/dA. As a first step towards reproducing Snell's law we equate that derivative with the sine of the angle :

\frac{dC_1}{dA_1} = \sin \alpha_1  \qquad \frac{dC_2}{dA_2} = \sin \alpha_2
(4)

Repeating the statement that is equivalent to Snell's law:

\frac{\sin\alpha_2}{\sin\alpha_1} = \frac{v_2}{v_1}
(5)

In order to accomodate the division by the velocity: let T1 be the time that it takes to traverse the length C1, T2 the time to traverse C2.

\frac{C_1}{v_1} = T_1 \qquad \frac{C_2}{v_2} = T_2
(6)

Combining:

\frac{dT_1}{dA_1} = \sin \alpha_1  \qquad \frac{dT_2}{dA_2} = \sin \alpha_2
(7)

Hence, in order to satisfy Snell's law:

\frac{dT_1}{dA_1} = \frac{dT_2}{dA_2}
(8)

A1 + A2 is constant, hence dA1 and -dA2 are equal. We can restate (8) as derivatives with respect to the variation (Snell's point S):

\frac{dT_1}{dS} = \frac{dT_2}{d(-S)}
(9)

Finally, we move the minus sign outside the differentiation:

(10)

(10) is our result: to identify the point in variation space that satisfies Snell's law we must identify the point where the derivative of T1 is equal to minus the derivative of T2.

Corollary: it follows from (10) that at the point in variation space where Snell's law is satisfied the derivative of the sum of T1 and T2 arrives at zero.



Picture 4. Graphlet
Snell's point is where the derivatives of T1 and T2 have the same magnitude.

Graphlet 4 displays the result: the curve labeled T1 represents the duration for the light to move from the point of emission to Snell's point, the curve labeled T2 represents the duration for the light to move from Snell's point to the point of reception.

The value that counts is not the duration 'T', but the derivative of that duration, the derivative with respect to the position of the point of refraction 'S'. It's the derivative that counts because that derivative is the ratio that produces the angle of Snell's law.

In graphlet 4 the two numbers in the right side subpanel give the slope of the curves T1 and T2 respectively.

Mathematics of comparing derivatives

We have that at the point where Snell's law is satisfied an extremum occurs for the value of the function (T1 + T2)

The fact that an extremum occurs is not specific to optics, or even physics: it is a universal property of the mathematics of functions and their derivatives.

The demand is to find the point where dT1/dS + dT2/dS reaches the value zero. It follows mathematically that the function (T1 + T2) arrives at an extremum at that point. However, that extremum is of no significance; the derivative counts, not the function itself.



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Last time this page was modified: May 22 2022