# calculus document

May 11th, 2015
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The Calculus of VariationsM. BenderskyRevised, December 29, 2008These notes are partly based on a course given by Jesse Douglas.1Contents1 Introduction. Typical Problems52 Some Preliminary Results. Lemmas of the Calculus of Variations103 A First Necessary Condition for a Weak Relative Minimum: The Euler-LagrangeDifferential Equation154 Some Consequences of the Euler-Lagrange Equation. The Weierstrass-ErdmannCorner Conditions.205 Some Examples256 Extension of the Euler-Lagrange Equation to a Vector Function, Y(x)327 Eulers Condition for Problems in Parametric Form (Euler-Weierstrass Theory) 368 Some More Examples449 The first variation of an integral, I(t) = J[y(x, t)] =R x2 (t)y(x,t)x1 (t) f (x, y(x, t), x )dx; Application to transversality.5410 Fields of Extremals and Hilberts Invariant Integral.5911 The Necessary Conditions of Weierstrass and Legendre.6312 Conjugate Points,Focal Points, Envelope Theorems6913 Jacobis Necessary Condition for a Weak (or Strong) Minimum: GeometricDerivation7514 Review of Necessary Conditions, Preview of Sufficient Conditions.7815 More on Conjugate Points on Smooth Extremals.8216 The Imbedding Lemma.87217 The Fundamental Sufficiency Lemma.9118 Sufficient Conditions.9319 Some more examples.9620 The Second Variation. Other Proof of Legendres Condition.10021 Jacobis Differential Equation.10322 One Fixed, One Variable E

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