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Section 1.7 Other problems of the calculus of variations

It would be a mistake to infer that the category of questions to which the calculus of variations is devoted is exhausted even by the quite general problem proposed in the last section. We can vary the problem there described by seeking a minimizing curve among those joining a fixed point and a fixed curve or two fixed curves, instead of two fixed points, or in many other ways.

The famous old isoperimetric problem of the ancients was that of finding a simply closed curve of given length which incloses the largest area. The solution is a circle, though it is not any too easy to prove that this is so. Analytically the problem may be formulated as that of finding an arc with equations in the parametric form

\begin{equation*} x=x(t),\quad y=y(t)\quad (t_1\leq t\leq t_2) \end{equation*}

satisfying the conditions

\begin{equation*} x(t_1)=x(t_2),\quad y(t_1)=y(t_2) \end{equation*}

but not otherwise intersecting itself, giving the length integral

\begin{equation*} \int_{t_1}^{t_2} \sqrt{(x')^2+(y')^2}\, dt \end{equation*}

a fixed value \(l\text{,}\) and maximizing the area integral

\begin{equation*} \frac{1}{2}\int_{t_1}^{t_2} (xy' - x'y)\, dt. \end{equation*}

The problems of the calculus of variations for which one or more integrals are to be given fixed values, while another is to be made a minimum or maximum, are called, after this one, isoperimetric problems. The problem proposed by James Bernoulli in 1697 was the earliest isoperimetric problem after that of the ancient Greeks.

It will not be possible for us in the limited space of the following pages to examine in detail more than the simpler non-isoperimetric problems, though there are many other types besides those which have already been mentioned.

The theory of the calculus of variations has been extensively developed but not so widely applied to special cases, very few of the particular problems having been exhaustively investigated. In the following Chapters II-IV three of the special problems mentioned in the preceding pages which have been studied in detail will be discussed, and in Chapter V some of the results for the more general problem formulated in Section 6 are collected, with a brief historical sketch of the progress of the theory from the time of the Bernoullis to the present.