Last update, Aug. 30, 2003

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General references on Calculus of Variations :

• W. S. Kimball :
  • Calculus of Variations - by Parallel Displacement, 1952

BibTeX references .

0. Introduction

1. The Fundamental Identities of the Calculus of Variations and Their Significance

2. The Vector Integrand and its Components for any Line Integral in a Plane

3. Dependent Line Integrals and the E-Function

4. Basic Equations for Solving all Maximum and Minimum Problems in the Calculus of Variations

5. Operational Technique and Applications

Geodesics.

Fermat's Principle (p.117):

• The path of a ray of light between two end-points is such as to minimize the time of transit between them.

Malus' Theorem (p.119):

• The optical path between any two wave front is the same for any ray.

This is equivalent to Fermat's.

6. Mechanics and the Calculus of Variations

7. Hilbert Integrals, Area Derivatives and the Legendre and Weirstrass Criteria for Extrema in the Calculus of Variations

8. The Envelope Theorem, Conjugate Points and Jacobi's Necessary Condition

Envelope Theorem (p. 292)

• Relates evolutes to single paths in the calculus of variations. Proved in the general case by Darboux and Zermelo (1894) and Kneser (1898). It states: "When a single parameter family of external paths from a fixed point O has an envelope, the integral from the fixed point to any point A on the envelope equals the integral from the fixed point to any second point B on the envelope plus the integral along the envelope to the first point on the envelope, J_OA = J_OB + J_BA ."

9. The Brachistochrone

10. Newton's Problem

11. Restricted Corner Conditions, the Vanishing E-Function and Variable End-Points in the Calculus of Variations

12. Unrestricted Corner Conditions and the Weirstrass-Erdmann Corner Condition

13. Minimum Surfaces of Revolution

14. The Series of Derived Hilbert Integrals in the Role of Families of Level Surfaces of the Transversal, Extremal and Equal Action Types: The Criterion for an Extremum of an Integral to be Minimum or Maximum

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