lecture04: The 1st Variation: Euler-Lagrange formulation of the fixed end-point problem

Topics covered

• The Catenary
• Fixed end-point variational problem
• Formulation
• The Catenary
• How do we tackle these problems
• Perturbations of functions
• The Functional of interest.
• Perturbation functions
• What to do
• The first variation
• Analogy to functions
• Simplification
• Euler-Lagrange equation
• A useful lemma
• A second useful lemma
• Proof of Euler-Lagrange equation
• Example: geodesics in a plane
• $f$ depends only on $y'$
• Example $f$ depends only on $y'$
• Fermat's principle
• Speed of light
• Example
• Example (ii)
• Example (iii)
• What we can't do (yet)
• How we might solve
• The functional
• The total time taken
• The result
• More than one boundary
• Dealing with ``kinks'' %
• Bernoulli and the Brachystochrone