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Examples and Exercises on Classical Problems

  • Lamberto Cesari
Part of the Applications of Mathematics book series (SMAP, volume 17)

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.

Keywords

Euler Equation Classical Problem Absolute Minimum Linear Manifold Polygonal Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  • Lamberto Cesari
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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