Part I Calculus of variations

1 The brachistochrone

When modern calculus was being developed during the seventeenth century, the Swiss mathematician Johan Bernoulli publicly posed a challenge problem for his peers to solve [4, Chapter 7]:

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

This situation is illustrated in Figure 1.1. The story goes that Bernoulli already knew the solution to this problem and taunted his rivals (with thinly veiled references, in particular, to Issac Newton) that they would not be able to solve it. The counterstory from Newton's camp is that he solved it easily after he got home from his day-job of managing "The Great Recoinage of 1696" (replacing one set of coins in Britain with another) as Warden of the Royal Mint.

Figure 1.1 Brachistochrone.

This problem is called the "brachistochrone" problem from the Greek words "brachistos" (the shortest) and "chronos" (time), and it was solved by other mathematicians in addition to Isaac Newton and Johann Bernoulli. The brachistochrone problem soon led to other problems of its type, and eventually-via Leonhard Euler and Joseph-Louis de Lagrange-to a new field of mathematics called "Calculus of Variations".

1.1 Calculus of variations formulation

The brachistochrone problem can be restated as finding the function x(t) whose graph not only goes through the two points A=(0,1) and B=(1,0) as in Figure 1.2, but also gives the path of least time.

Figure 1.2 Brachistochrone on standard axes.

The time it takes for a point to move via gravity along the graph of any such x(t) can be obtained by integrating the inverse of the speed with respect to arclength, which in this case simplifies to a constant multiple of the integral "functional" J[x] defined by

in terms of the derivative x?=x?(t) with respect to t of the function x=x(t).

1.1.1 Functionals

Recall that functions take input variables (points) and generate output values (numbers). The "al" at the end of the word functional indicates that the input variables are functions x(t) themselves, and we use the square brackets " [x]" to signal this distinction. Thus the integral J[x] defined in (1.1) is a function-like object that generates an output value (i.?e., the value of the integral) from any function x(t) to which it is applied.

Problem 1.1.

(a)

Simplify the output values J[x?] for the integral functional (1.1) applied to functions of the form

x?(t)=1-t?for?>0

as much as possible (without computing the integral).

(b)

Compute the output value J[x1] and describe the graph of x1(t).

1.1.2 Admissibility

Since the integral functional J[x] defined in (1.1) measures (a constant multiple of) the time it takes for a point to move via gravity along the graph of x(t), it follows that the brachistochrone problem amounts to minimizing J[x] over functions x(t) satisfying x(0)=1 and x(1)=0 (ensuring that the graph of x(t) connects the points A=(0,1) and B=(1,0)). The term admissible is used to identify the functions that we consider for the minimization, so the admissible functions x(t) for the brachistochrone problem must at least satisfy x(0)=1 and x(1)=0.

The original statement of the brachistochrone problem said nothing about functions x(t). However, for a point to move via gravity from A to B in the shortest time, it is clear that it must follow the graph of a continuous function. Our shorthand notation for this property of the function x(t) is x?C0. We also use the shorthand notation x?C1 to indicate that x(t) is differentiable with continuous derivative x?(t), and x?C2 to indicate that x(t) is twice-differentiable with continuous first and second derivatives x?(t) and x¨(t):

Notice that our statement of the brachistochrone problem via the integral functional J[x] defined in (1.1) implicitly assumes that we can compute the corresponding derivatives x?(t). Since we can always break the integral at a finite number of discontinuities, we only need the derivatives x?(t) to be piecewise continuous on [0,1], which means that there are at most a finite number of inputs t?[0,1] at which the derivative function x?(t) is not continuous. This leads us to one last shorthand notation:

It follows that our version of the brachistochrone problem can be stated compactly as

The admissible functions in (1.2) are the x?D1 that satisfy the endpoint conditions x(0)=1 and x(1)=0.

Problem 1.2.

(a)

It should be clear that the function-categories satisfy the relationships

C2?C1?C0,

so that, for instance, a function x?C2 is necessarily in the other two categories. How does the category D1 fit in this scheme?

(b)

Sketch the graph of one example of a D1 function that is not a C1 function?

1.1.3 Solution

To solve the brachistochrone problem (1.2), we will learn how to minimize integral functionals. The variables in this case are functions x(t) themselves, and in the following chapter, we will see how to construct "variations" of a function to use for comparison in the minimization. These objects give the subject "calculus of variations" part of its name, and their development was motivated by the brachistochrone problem and other problems like it proposed in that era. We can use calculus of variations to show in particular that the solution to the brachistochrone problem (1.2) is a piece of a "cycloid". Figure 1.3 illustrates how a cycloid can be generated by rolling a circle from left to right along a horizontal axis.

Figure 1.3 Cycloid generated by rolling circle.

Restricted admissibility [3, adapted...