Variational Calculus

Variations

The calculus of variations is concerned with the extremals of functions whose domain is an infinite-dimensional space: the space of curves. Such functions are called functionals.

Differentiable Functional

A functional on the space of functions is called differentiable if , such that depends linearly on and is a remainder term that in the sense that for and , we have for some positive constant . is called the differential or variation of at , denoted as .

Proposition

Suppose is a differentiable functional of curves, and is a (not necessarily linear) map of coordinate change. If we identify as a transformation that change the coordinate of every point on a curve, then the functional after changing the coordinate, , is also a differentiable functional on .

Proof Fix some arbitrary curve , for any and any , since is differentiable, we can write where is the differential of at , and is a remainder term that satisfies will imply for some constant . As the above holds for all , we shall abuse our notation and identify , and to be the curve in such that Hence the above equation can be rewritten without :Now we consider the differentiable functional at , for which we denote be the variation of at . Then we have the following equation hold: To simplify our notation, let , then we have where the first equation is by differentiability of , the second equation is by the linearity of . Without loss of generality, pick some small , suppose and , we have for some constant . Since is linear, we have for some positive constant . Moreover, is also linear, so for some positive constant . Similarly, satisfies: because is bounded by , is bounded by some , and is bounded by . Set , then both and are bounded by . Therefore, there holds Thus, which means , so is differentiable, and .

Proposition

Suppose . The functional is differentiable, and its derivative is given by

Proof This is achieved by taking Taylor expansion, and then integral by parts.

Euler-Lagrange Equation

Extremal

An extremal of a differentiable functional is a curve such that .

Lemma

If a continuous real function defined on satisfies for all with , then .

Proof We prove by contradiction. Suppose for some for some . Since is continuous, in some open ball of the point . Let be a smooth function such that in , outside of , and in . Then, clearly, . This yields a contradiction, so for all .

Euler-Lagrange Equation

Suppose . The curve is an extremal of the functional on the space of smooth curves passing through the points and if and only if it satisfies This is called the Euler-Lagrange equation for the functional.

Proof By the preceding proposition, we know Since is defined on the space of smooth curves passing through points and , we require , thus the second term vanishes. Therefore, by definition, the extremal is achieved when By the above lemma, it follows that at all time. The converse is true trivially.

Theorem

The condition for a curve to be an extremal of a functional does not depend on the coordinate system.

Proof Suppose we have two coordinate systems and to describe same curve . Then, the Euler-Lagrange equation in coordinates is Note that is a coordinate transformed from , so . Thus , so . Therefore the above equation can be written as Clearly is non-zero, so , which is the Euler-Lagrange equation in coordinates.

Remark

This is an important point, because it helps us get coordinate independent descriptions of mechanical systems.

Beltrami Identity

The Beltrami identity for some function is defined as

Proposition

For any , we have the Euler-Lagrange Equation is equivalent to .

Here is one example of variational calculus in optics:

Fermat’s Principle

Fermat’s Principle states that the path taken by a light ray between two points is the one that requires the least time. Mathematically, this principle can be expressed as , where is the index of refraction and is the distance traveled by the light ray.