Calculus of Variations: Simple Definition

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Calculus of variations is, loosely speaking, the study of extreme values—the optimization of certain quantities over sets of functions. More technically, it’s the mathematical analysis of nonlinear minimization principles on infinite-dimensional function spaces (Olver, 2021). The field can also be described as a tool where partial differentiation equations can be reformulated as minimization problems.

Calculus of variations can answer questions like: what is the shortest line that connects point A to point B? The solution isn’t always a straight line. For example, let’s say a lifeguard (at point A) is trying to reach a surfer in the ocean (at point B). The lifeguard’s speed on the sand is greater than their speed in the water, which means the optimal line is made up of two line segments: one on the sand, and one in the water. This optimal path satisfies Snell’s law, represented by the equation:

Ordinary Calculus vs. Calculus of Variations

With “ordinary” calculus, the focus is on real functions, which map real numbers to real numbers. For example, the cosine function maps the real number line to the closed interval [-1, 1]. With calculus of variations, the focus shifts to functionals: objects that map functions, instead of numbers, to the real number line. More generally speaking, the quantity S[g] is a functional of g(x) in the interval [a, b] if it depends on every value of f(x) in [a, b].

For example, the functional:


maps the real function g to a real number S[g] (OU, 2016).

References

Olver, P. (2021). The Calculus of Variations. Retrieved February 1, 2021 from: https://www-users.math.umn.edu/~olver/ln_/cv.pdf
OU (2016). Introduction to the calculus of variations. Retrieved February 1, 2021 from: https://www.open.edu/openlearn/ocw/pluginfile.php/1118521/mod_resource/content/3/Introduction%20to%20the%20calculus%20of%20variations_ms327.pdf

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