The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions; Convexity and Lower Semicontinuity

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


We are concerned here with classical Lagrange problems of the calculus of variations (see Section 1.1). Precisely, we are concerned with minima and maxima of the functional
$$ I\left[ x \right] = \int_{{{{t}_{1}}}}^{{{{t}_{2}}}} {{{f}_{0}}(t,x(t),x'(t))dt,x(t) = ({{x}^{1}}, \ldots ,{{x}^{n}})} , $$
with constraints and boundary conditions
$$\left( {t,x\left( t \right)} \right) \in A,{\text{ t}} \in \left[ {{t_1},{t_2}} \right],{\text{ }}\left( {{t_1},x\left( {{t_1}} \right),{t_2},x\left( {{t_2}} \right)} \right) \in B$$


Euler Equation Absolute Minimum Absolutely Continuous Canonical Equation Invariant Character 
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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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