Examples and Exercises on Classical Problems

Abstract

Let us briefly consider, in terms of the integral ${\text{I}} = {{\int_{{{{t}_{1}}}}^{{{{t}_{2}}}} {\left( {1 + {{{x'}}^{2}}} \right)} }^{{1/2}}}dt,$, the question of the path of minimum length between two fixed points, or between a fixed point and a given curve in the tx-plane. Here ${{f}_{0}} = {{\left( {1 + {{{x'}}^{2}}} \right)}^{{1/2}}}$ depends on x’ only, and satisfies condition (S) of Section 2.8. Any extremal must satisfy Euler’s equation in the reduced form (2.2.10), ${{f}_{{0x'}}} = c,$ or $x'{{\left( {1 + {{{x'}}^{2}}} \right)}^{{ - 1/2}}} = c$. Here ${{f}_{{0x'}}} = x'{{\left( {1 + {{{x'}}^{2}}} \right)}^{{ - 1/2}}}$ is a strictly increasing function of x′ with range (−1,1). Thus −1 < c < 1, and there is one and only one value x′ = m, depending on c, such that $x{\left( {1 + {{x}^2}} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} = c$. From (2.6.iii) we know that any optimal solution is of class C² and therefore an extremal. Thus an optimal solution must be a segment x′ = m, x(t)= mt + b,m, b constants.