The Calculus of Variations

The following sections are included:

• Foundations of the Calculus of Variations

  • A Simple Minimization Problem

  • Methods of the Calculus of Variations

    • Euler's First Equation

    • Extremal Values of an Integral

    • Jacobi Equation^*

    • Euler's Second Equation

  • Path of Shortest Distance and Geodesic Equation

    • Geodesic equation^*

    • Geodesic equation on a sphere

• Classical Variational Problems

  • Isoperimetric Problem

  • Brachistochrone Problem

• Fermat's Principle of Least Time

  • Light Propagation in a Nonuniform Medium

  • Snell's Law

  • Application of Fermat's Principle

• Geometric Formulation of Ray Optics^*

  • Frenet-Serret Curvature of Light Path

  • Light Propagation in Spherical Geometry

  • Geodesic Representation of Light Propagation

    • Vacuum-metric case

    • Medium-metric case

    • Jacobi equation for light propagation

  • Wavefront Representation

• Problems