Section 2.1 The shortest arc joining two points
¶Problems of determining shortest distances furnish a useful elementary introduction to the theory of the calculus of variations because the properties characterizing their solutions are familiar ones which illustrate very well many of the general principles common to all of the problems suggested in the preceding chapter. If we can for the moment eradicate from our minds all that we know about straight lines and shortest distances we shall have the pleasure of rediscovering well-known theorems by methods which will be helpful in solving more complicated problems.
Let us begin with the simplest case of all, the problem of determining the shortest arc joining two given points. The integral to be minimized, which we have already seen in Section 1.3, may be written in the form
if we use the notation \(f(y')=\sqrt{1+(y')^2}\text{,}\) and the arcs \(y=y(x)\,\,(x_1\leq x\leq x_2)\) whose lengths are to be compared with each other will always be understood to be continuous and to consist of a finite number of arcs on each of which the tangent turns continuously, as indicated in Figure 2.1.1.
Analytically this means that on the interval \(x_1\leq x\leq x_2\) the function \(y(x)\) is continuous and that the interval can be subdivided into parts on each of which \(y(x)\) has a continuous derivative. Let us agree to call such functions admissible functions and the arcs which they define admissible arcs. Our problem is then to find among all admissible arcs joining two given points 1 and 2 one which makes the integral \(I\) a minimum.
Activity 2.1.1. Admissible arcs, smoothness, and classes of functions.
The idea of smoothness, which generalizes differentiability, is one that comes up a lot in analysis. We can categorize functions into several different differentiability classes based on how many continuous derivatives they have over some domain.
The first two classes are called \(C^0\) and \(C^1\text{.}\) \(C^0\) consists of all the continuous functions, and \(C^1\) is the class of differentiable functions whose derivative is continuous. Give an example of a function in \(C^0\) that isn't in \(C^1\text{.}\)
You may have noticed that in the definition of admissible functions above, we're allowing some cusps, as long as the function is mostly differentiable. This condition is more commonly referred to as being piecewise differentiable or piecewise \(C^1\). Give an example of a function that's not in \(C^1\text{,}\) but is piecewise \(C^1\text{.}\)
The very nicest functions (and, tbh, most of the ones we care about in everyday life) have infintely many derivatives. These are called smooth functions, and they belong to the class \(C^\infty\text{.}\) Give three examples of functions in \(C^\infty\text{.}\)