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The Calculus of Variations M. Bendersky ∗ December 1, 2022 ∗These notes are partly based on a course given by Jesse Douglas. 1 Contents 1 Introduction. Typical Problems 5 2 Some Preliminary Results. Lemmas of the Calculus of Variations 9 3 AFirst Necessary Condition for a Weak Relative Minimum: The Euler- Lagrange Differential Equation 15 4 Some Consequences of the Euler-Lagrange Equation. The Weierstrass- Erdmann Corner Conditions. 20 5 Some Examples 24 6 Extension of the Euler-Lagrange Equation to a Vector Function, Y(x) 32 7 Euler’s Condition for Problems in Parametric Form (Euler-Weierstrass Theory) 36 8 Some More Examples 44 9 The first variation of an integral, I(t) = J[y(x,t)] = Rx (t) ∂y(x,t) 2 f(x,y(x,t), )dx; Application to transversality. 53 x1(t) ∂x 10Fields of Extremals and Hilbert’s Invariant Integral. 58 11The Necessary Conditions of Weierstrass and Legendre. 61 2 12Conjugate Points,Focal Points, Envelope Theorems 67 13Jacobi’s Necessary Condition for a Weak (or Strong) Minimum: Geo- metric Derivation 72 14Review of Necessary Conditions, Preview of Sufficient Conditions. 75 15More on Conjugate Points on Smooth Extremals. 79 16The Imbedding Lemma. 83 17The Fundamental Sufficiency Lemma. 87 18Sufficient Conditions. 89 19Some more examples. 91 20The Second Variation. Other Proof of Legendre’s Condition. 95 21Jacobi’s Differential Equation. 97 22One Fixed, One Variable End Point. 106 23Both End Points Variable 111 24Some Examples of Variational Problems with Variable End Points 114 25Multiple Integrals 118 3 26Functionals Involving Higher Derivatives 124 27Variational Problems with Constraints. 130 28Functionals of Vector Functions: Fields, Hilbert Integral, Transversality in Higher Dimensions. 146 29The Weierstrass and Legendre Conditions for n ≥ 2 Sufficient Conditions.160 30The Euler-Lagrange Equations in Canonical Form. 164 31Hamilton-Jacobi Theory 168 31.1 Field Integrals and the Hamilton-Jacobi Equation. . . . . . . . . . . . . . . . 168 31.2 Characteristic Curves and First Integrals . . . . . . . . . . . . . . . . . . . . 173 31.3 A theorem of Jacobi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 31.4 The Poisson Bracket. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 31.5 Examples of the use of Theorem (31.10) . . . . . . . . . . . . . . . . . . . . . 181 32Variational Principles of Mechanics. 183 33Further Topics: 186 4
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