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Calc of Variations Notes

# Calc of Variations Notes - The Calculus of Variations M...

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The Calculus of Variations M. Bendersky * Revised, December 29, 2008 * These notes are partly based on a course given by Jesse Douglas. 1

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Contents 1 Introduction. Typical Problems 5 2 Some Preliminary Results. Lemmas of the Calculus of Variations 10 3 A First 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 25 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 )] = R x 2 ( t ) x 1 ( t ) f ( x, y ( x, t ) , ∂y ( x,t ) ∂x ) dx ; Application to transversality. 54 10 Fields of Extremals and Hilbert’s Invariant Integral. 59 11 The Necessary Conditions of Weierstrass and Legendre. 63 12 Conjugate Points,Focal Points, Envelope Theorems 69 13 Jacobi’s Necessary Condition for a Weak (or Strong) Minimum: Geometric Derivation 75 14 Review of Necessary Conditions, Preview of Sufficient Conditions. 78 15 More on Conjugate Points on Smooth Extremals. 82 16 The Imbedding Lemma. 87 2
17 The Fundamental Sufficiency Lemma. 91 18 Sufficient Conditions. 93 19 Some more examples. 96 20 The Second Variation. Other Proof of Legendre’s Condition. 100 21 Jacobi’s Differential Equation. 103 22 One Fixed, One Variable End Point. 113 23 Both End Points Variable 119 24 Some Examples of Variational Problems with Variable End Points 122 25 Multiple Integrals 126 26 Functionals Involving Higher Derivatives 132 27 Variational Problems with Constraints. 138 28 Functionals of Vector Functions: Fields, Hilbert Integral, Transversality in Higher Dimensions. 155 29 The Weierstrass and Legendre Conditions for n 2 Sufficient Conditions. 169 30 The Euler-Lagrange Equations in Canonical Form. 173 31 Hamilton-Jacobi Theory 177 31.1 Field Integrals and the Hamilton-Jacobi Equation. . . . . . . . . . . . . . . . . . . . 177 31.2 Characteristic Curves and First Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 182 31.3 A theorem of Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 31.4 The Poisson Bracket. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 31.5 Examples of the use of Theorem (31.10) . . . . . . . . . . . . . . . . . . . . . . . . . 190 32 Variational Principles of Mechanics. 192 3

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33 Further Topics: 195 4
1 Introduction. Typical Problems The Calculus of Variations is concerned with solving Extremal Problems for a Func- tional . That is to say Maximum and Minimum problems for functions whose domain con- tains functions, Y ( x ) (or Y ( x 1 , · · · x 2 ), or n -tuples of functions). The range of the functional will be the real numbers, R Examples : I. Given two points P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) in the plane, joined by a curve, y = f ( x ). The Length Functional is given by L 1 , 2 ( y ) = R x 2 x 1 p 1 + ( y 0 ) 2 dx | {z } ds . The domain is the set of all curves, y ( x ) C 1 such that y ( x i ) = y i , i = 1 , 2. The minimum problem for L [ y ] is solved by the straight line segment P 1 P 2 . II. (Generalizing I) The problem of Geodesics , (or the shortest curve between two given points) on a given surface. e.g. on the 2-sphere they are the shorter arcs of great circles (On the Ellipsoid Jacobi (1837) found geodesics using elliptical co¨ o rdinates in terms of Hyperelliptic integrals, i.e.

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