Calculus of Variations
Fermat's Principle
Fermats principle is that light 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]
- [math]= \frac{n_1}{c}\int_A^I ds+ \frac{n_2}{c} \int_I^B ds [/math]
- [math]= \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]x[/math] to find a minimum for the time of flight
- [math] \frac{d t}{dx} = 0[/math]
- [math] \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] = \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] = \frac{n_1}{c}\frac{x}{\sqrt{ h_1^2 + x^2 }} + \frac{n_2}{c} \frac{(\ell -x)(-1)}{\sqrt{h_2^2 + (\ell -x)^2}} [/math]
- [math] = n_1\frac{x}{\sqrt{ h_1^2 + x^2 }} - n_2 \frac{\ell -x}{\sqrt{h_2^2 + (\ell -x)^2}} [/math]
- [math] \Rightarrow n_1\frac{x}{\sqrt{ h_1^2 + x^2 }} = n_2 \frac{\ell -x}{\sqrt{h_2^2 + (\ell -x)^2}} [/math]
or
- [math] \Rightarrow n_1\sin(\theta_1) = n_2 \sin(\theta_2) [/math]
Generalizing Fermat's principle to determining the shorest path
One can apply Fermat's principle to show that the shortest path between two points is a straight line.
In 2-D one can write the differential path length as
- [math]ds=\sqrt{dx^2+dy^2}[/math]
using chain rule
- [math]dy = \frac{dy}{dx} dx \equiv y^{\prime}(x) dx[/math]
the the path length between two points [math](x_1,y_1)[/math] and [math](x_2,y_2)[/math] is
- [math]S = \int_{(x_1,y_1)}^{(x_2,y_2)} ds= \int_{(x_1,y_1)}^{(x_2,y_2)} \sqrt{dx^2+dy^2}[/math]
- [math] = \int_{(x_1,y_1)}^{(x_2,y_2)} \sqrt{dx^2+\left ( y^{\prime}(x) dx\right)^2}[/math]
- [math] = \int_{(x_1,y_1)}^{(x_2,y_2)} \sqrt{1+y^{\prime}(x)^2}dx[/math]
adding up the minimum of the integrand function is one way to minimize the integral ( or path length)
let
- [math]f(y,y^{\prime},x) \equiv \sqrt{1+y^{\prime}(x)^2}[/math]
the path integral can now be written in terms of dx such that
- [math]S= \int_{(x_1)}^{(x_2)} f(y,y^{\prime},x) dx[/math]
To consider deviation away from [math]y(x)[/math] the function [math]\eta(x)[/math] is introduced to denote deviations away from the shortest line and the parameter [math]\alpha[/math] is introduced to weight that deviation
- [math]\eta(x)[/math] = the difference between the current curve and the shortest path.
let
- [math]Y(x) = y(x) + \alpha \eta(x)[/math] = A path that is not the shortest path between two points.
let
- [math]S(\alpha)= \int_{(x_1)}^{(x_2)} f(Y,Y^{\prime},x) dx[/math]
- [math]= \int_{(x_1)}^{(x_2)} f(y+\alpha \eta,y^{\prime}+ \alpha \eta^{\prime},x) dx[/math]
To find the critical point we need to differentiate that above integral with respect to the parameter [math]\alpha[/math] as this parameter changes the deviation
- [math]\frac{\partial}{\partial \alpha} f(y+\alpha \eta,y^{\prime}+ \alpha \eta^{\prime},x) = \eta \frac{\partial f}{\partial y}+\eta^{\prime} \frac{\partial f}{\partial y^{\prime}}[/math]
a critical point (where a min or max) exists if
- [math]\frac{df}{dx} = 0[/math]
if the second derivative is zero there is a min, max, or neither
if the second derivative is negative there is a max
if the second derivative is positive there is a min
- [math]\frac{df}{dx} = \frac{d}{dx} \left ( y^{\prime} \frac{\partial f}{\partial y^{\prime}} \right )[/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
