Forest UCM CoV

Calculus of Variations

Fermat's Principle

Fermats principle is thatlight takes a path between two points that requires the least amount of time.

If we let S represent the path of light between two points then

[math]S=vt[/math]

light takes the time [math]t[/math] to travel between two points can be expressed as

[math]t = \int_A^B dt =\int_A^B \frac{1}{v} ds [/math]

The index of refraction is denoted as

[math]n=\frac{c}{v}[/math]

[math]t = \int_A^B \frac{n}{c} ds [/math]

for light traversing an interface with an nindex of refraction $n_1$ on one side and $n_2$ on the other side we would hav e

[math]t = \int_A^I \frac{n_1}{c} ds+ \int_I^B \frac{n_2}{c} ds [/math]

http://scipp.ucsc.edu/~haber/ph5B/fermat09.pdf

Euler-Lagrange Equation

https://www.fields.utoronto.ca/programs/scientific/12-13/Marsden/FieldsSS2-FinalSlidesJuly2012.pdf

Forest_Ugrad_ClassicalMechanics