Section 26–3 Fermat’s principle of least time

(Law of reflection / Law of refraction / Infinitesimal change)

 

In this section, Feynman explains the laws of reflection and refraction through Fermat’s principle of least time, showing that the correct path is the one for which the first-order infinitesimal change in travel time vanishes. A more accurate title could be the principle of stationary time, since the path light follows is one where the travel time is stationary — meaning that small variations give no first-order change. This stationary value could correspond to a minimum, a maximum, or even a saddle point, though in optics it usually manifests as a minimum.

 

1. Law of reflection:

“…the angle of incidence equals the angle of reflection is equivalent to the statement that the light goes to the mirror in such a way that it comes back to the point B in the least possible time (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

According to Feynman, Fermat formulated the principle of least time around 1650 as: “that out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.” In a 1657 letter to Cureau de la Chambre, Fermat suggested that the law of refraction could be deduced from such a minimum principle. The principle can be expressed as: a light ray traveling between two points follows the path of shortest optical path length. The optical path length, nd, is the distance in vacuum that light would cover in the same time it traverses a distance d in a medium of refractive index n. In other words, the path of least time corresponds to the path with the minimum number of wavelengths.

 

“The first way of thinking that made the law about the behavior of light evident was discovered by Fermat in about 1650, and it is called the principle of least time, or Fermat’s principle (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Feynman states that the first way of thinking which made the laws of light evident was discovered by Fermat. However, as early as ~100 AD, Hero of Alexandria proposed that light reflects from a plane mirror by taking the path of shortest distance. Later, around ~200 AD, Damianus suggested an idea closer to the principle of least time. As Mach (1953) notes: “Damianus says ‘… the nature of our ray of vision did not permit of aimless wandering, the ray would be bent (reflected) at equal angles.’ As already mentioned, it is not quite clear whether the teleological conception refers to a minimum of time, or space, or perhaps both” (p. 28). Ultimately, Fermat’s contribution stands out because he formulated the principle in a mathematically precise way and extended it to refraction, where the shortest path is no longer valid.

 

2. Law of refraction:

“…let us demonstrate that the principle of least time will give Snell’s law of refraction (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Using the principle of least time to explain Snell’s law, Feynman (1985) illustrates the problem with a vivid analogy: a beautiful girl has fallen out of a boat and is crying for help in the water. In his lecture on QED for the public, he rephrases the problem as: “Suppose you’re the lifeguard, sitting at S, and the beautiful girl is drowning at D” (p. 51). He then distinguishes four kinds of path: (1) the path of least time, (2) the path of least water, (3) the path of least distance, and (4) the path certainly not of least time. To clarify why the least-time path is optimal, Feynman adds that “light seems to go slower in water than it does in air, which makes the distance through water more ‘costly,’ so to speak, than the distance through air” (p. 51). The word costly is fitting, since Fermat described his law as the principle of natural economy.

Source: (Feynman, 1985)

Recently, researchers have discovered light behaviors that go beyond Fermat’s principle. For example, a new class of laser beams called spacetime wave packets that do not follow the traditional law of refraction. As Bhaduri et al. (2020) observe, “space-time refraction defies our expectations derived from Fermat’s principle.” This reminds us that Fermat’s principle should not be treated as fundamental, nor should we assume that all light beams necessarily follow the shortest-time path. In fact, light can obey different rules depending on the experiment, and its behavior is more comprehensively explained by the broader framework of the principle of stationary action. Strictly speaking, the phrase “least action” is a misnomer: the true path corresponds to a stationary value of the action, which may represent a minimum, a maximum, or even a saddle point (Zengel, 2024).

 

3. Infinitesimal change:

“This means that if we move the point X to points near C, in the first approximation there is essentially no change in time because the slope is zero at the bottom of the curve. So our way of finding the law will be to consider that we move the place by a very small amount, and to demand that there be essentially no change in time (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

Feynman explains that if we compare the travel times at points near C, there is essentially no change in time to first order, because the slope is zero at the bottom of the curve (Fig. 26–5). In other words, optical laws can be deduced by comparing the times at neighboring points that differ only by an infinitesimal displacement. To see what “first approximation” means more precisely, we can use Taylor’s theorem: f(x) = f(a) + (x−a)f′(a)/1! + (x–a)²f′′(a)/2! + … (x – a)ⁿfⁿ(a)/n! + … The zeroth-order approximation is simply the constant term f(a). The first-order approximation adds the linear term (x−a)f′(a), which locally replaces the curve with its tangent line at x=a. In other words, the curve is approximated by using a straight line in the immediate neighborhood of the point.

“Of course, there is an infinitesimal change of a second order; we ought to have a positive increase for displacements in either direction from C (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Feynman remarks that the infinitesimal change near point C is of second order, meaning that for small displacements to either side of C, the travel time increases only quadratically (Fig. 26–5). To clarify this idea, we can turn to the second-order term of Taylor’s expansion, (x–a)²f′′(a)/2!, which represents a quadratic approximation to the curve. In other words, while the first-order approximation replaces the curve with a straight line, the second-order approximation uses a parabola. As Feynman later emphasizes in The Feynman Lectures on Physics, Vol. II (1964), “one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order.” Put differently, the second-order effect, by contrast, is governed by the second derivative of the function, which reflects the curvature of the curve at the minimum.

 

Review Questions:

1. Feynman credits Fermat with the first clear way of thinking about the behavior of light. Do you agree, or do earlier thinkers like Hero of Alexandria and Damianus deserve more recognition?

2. How would you explain the law of refraction using Fermat’s principle of least time?

3. The idea of an infinitesimal change of first order plays a key role in Fermat’s principle. How would you interpret this mathematically and conceptually?

 

Key Takeaway: We can derive optical laws (such as the laws of reflection and refraction) by analyzing how the first-order change in travel time vanishes (stationary point) for different possible paths of light.

 

The Moral of the Lesson:

The Taoist concept of Wúwéi, often translated as "non-action" or "effortless action," does not imply passivity or doing nothing. Rather, it is the art of achieving maximum effect with minimal, economical, or natural effort—of moving in harmony with the inherent patterns of nature, like following the path of least resistance. A wise president who practices Wúwéi governs not through forceful control, but by aligning with the natural order—just as a physical system follows the principle of stationary action without the burden of unnecessary constraints.

 

References:

Bhaduri, B., Yessenov, M., & Abouraddy, A. F. (2020). Anomalous refraction of optical spacetime wave packets. Nature Photonics, 14, 416-421.

Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

Mach, E. (1953). The Principles of Physical Optics: An Historical and Philosophical Treatment. New York: Dover.

Zengel, K. (2024). Why the action?. American Journal of Physics, 92(11), 885-888.