The Calculus of Variations
                                                     M. Bendersky ∗
                                                    December 1, 2022
                   ∗These notes are partly based on a course given by Jesse Douglas.
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                   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
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               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
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                    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
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