Calc of Variations Notes

Calc of Variations - The Calculus of Variations M Bendersky Revised These notes are partly based on a course given by Jesse Douglas 1 Contents 1

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The Calculus of Variations M. Bendersky * Revised, December 29, 2008 * 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 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 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 )....
View Full Document

This note was uploaded on 10/28/2011 for the course MATH 768 taught by Professor Berndesky during the Spring '11 term at CUNY Hunter.

Page1 / 196

Calc of Variations - The Calculus of Variations M Bendersky Revised These notes are partly based on a course given by Jesse Douglas 1 Contents 1

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online