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]\displaystyle{ S=vt }[/math]
light takes the time [math]\displaystyle{ t }[/math] to travel between two points can be expressed as
- [math]\displaystyle{ t = \int_A^B dt =\int_A^B \frac{1}{v} ds }[/math]
The index of refraction is denoted as
- [math]\displaystyle{ n=\frac{c}{v} }[/math]
- [math]\displaystyle{ 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]\displaystyle{ t = \int_A^I \frac{n_1}{c} ds+ \int_I^B \frac{n_2}{c} ds }[/math]
- [math]\displaystyle{ = \frac{n_1}{c}\int_A^I ds+ \frac{n_2}{c} \int_I^B ds }[/math]
- [math]\displaystyle{ = \frac{n_1}{c}\sqrt{h_1^2 + x^2}+ \frac{n_2}{c} \sqrt{h_2^2 + (\ell -x)^2} }[/math]
take derivative of time with respect to [math]\displaystyle{ x }[/math] to find a minimum for the time of flight
- [math]\displaystyle{ \frac{d t}{dx} = 0 }[/math]
- [math]\displaystyle{ \Rightarrow 0 = \frac{d}{dx} \left ( \frac{n_1}{c}\left ( h_1^2 + x^2 \right )^{\frac{1}{2}}+ \frac{n_2}{c} \left (h_2^2 + (\ell -x)^2 \right)^{\frac{1}{2}} \right ) }[/math]
- [math]\displaystyle{ = \frac{n_1}{c}\left( h_1^2 + x^2 \right )^{\frac{-1}{2}} (2x) + \frac{n_2}{c} \left (h_2^2 + (\ell -x)^2 \right)^{\frac{-1}{2}} 2(\ell -x)(-1) }[/math]
- [math]\displaystyle{ = \frac{n_1}{c}\frac{2x}{\sqrt{ h_1^2 + x^2 }} + \frac{n_2}{c} \frac{(\ell -x)(-1)}{\sqrt{h_2^2 + (\ell -x)^2}} }[/math]
- [math]\displaystyle{ = n_1\frac{2x}{\sqrt{ h_1^2 + x^2 }} - n_2 \frac{\ell -x}{\sqrt{h_2^2 + (\ell -x)^2}} }[/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