1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 11
G. A. Krafft
Jefferson Lab

2. Calculus of Variations
General method for finding functions that extremize integrals
Solution by Euler-Lagrange equation
Newton’s Law of motion is an example
Equivalent to Hamilton’s “Principle of Least Action”
Equations of motion can be written and solved in Euler-Lagrange form
Euler-Lagrange equations apply in arbitrary coordinate systems, including accelerating or rotating coordinate systems, in contrast to F=ma which only applies in inertial systems.
Easy identification of constants of motion

3. Foundational Result
Finding extreme values of functions is easily accomplished by standard methods in calculus. Suppose instead we wish to find the function y(x) which makes the integral
take on extreme values, either maximum or minimum. Our goal, find a necessary (but not sufficient!) condition that y(x) must satisfy when the integral extremal.
Suppose g(x) is continuous and for all h(x) differentiable with h(a) = h(b) =0,
vanishes. What can we say about g(x)?

4. Suppose g(x)≠0 somewhere, call the location x0
Suppose g(x)≠0 somewhere, call the location x0. By continuity, there is a small band [x1,x2] containing x0 where g(x)>g(x0)/2 if g(x0) is positive or g(x)<g(x0)/2 if g(x0) is negative. Consider the function h(x) = sign[g(x0)](x-x1)2(x-x2)2 for x inside [x1,x2] and vanishing outside [x1,x2] . It is clearly strictly non-negative or non-positive, differentiable, and has a non-zero integral. Call its integral over the closed interval I[x1,x2]. But then
contradicting the hypothesis.
Therefore, g(x) must vanish throughout [a,b] .

5. Euler-Lagrange Equation
y(x)+δy(x) (wrong) a b

6. Example: Shortest Distance
“The shortest distance between two points is a straight line”
Argument entirely symmetrical under the interchange x ↔ y. Euler-Lagrange then yields

7. Classical Brachistochrone Problem
What is the curve x(y) between two points that provides the shortest time to descend, assuming the velocity is zero at the highest point?
Euler-Lagrange equation gives
(0,0)
(x1,y1)

8. Cycloid Solution Trigonometric Substitution Solution for x(θ)

9. 6.25 Cycloid Equal Time Property

10. Following the problem hint
Remarkable fact: does not depend on θ0!

11. Two Dimensional Solution
Can also do two dimensions separately as a function of time
Euler-Lagrange

12. Using the first Euler-Lagrange Equation
Or using the second
Form of solution does not depend on coordinates used!

13. Fermat’s Principle of Least Time
The travel time of a light ray, moving from point A to point B, is minimum along the actual path
In general, velocity of light depends on index of refraction n = c/v (>1) which can be a function of position n(x,y,z)
If path is in a plane (will happen if the index of refraction does not vary in some direction), the problem is to minimize the quantity
Strictly speaking, the path is stationary, e.g. reflected rays are not on distance minima

14. Snell’s Law 1: Problem 6.4
Actual light path has least time

15. Snell’s Law 2
Suppose used θ1 as independent variable instead