The Euler-Lagrange Equations

  • John L. Troutman
Part of the Undergraduate Texts in Mathematics book series (UTM)


Jakob Bernoulli’s solution of 1696 to his brother Johann’s problem of the brachistochrone (§1.2) marked the introduction of variational considerations. However, it was not until the work of Euler (c. 1742) and Lagrange (1755) that the systematic theory now known as the calculus of variations emerged. Initially, it was restricted to finding conditions which were necessary in order that an integral function
$$ F(Y) = \int_a^b {f(x,y(x),y'(x))dx = } \int_a^b {f[y(x)]dx} $$
should have a (local) extremum on a set D ⊆ {yC1[a,b]: y(a) = a1;y(b) = b1}.


Lagrangian Multiplier Stationary Function Transversal Condition Extremal Function Natural Boundary Condition 
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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • John L. Troutman
    • 1
  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA

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