1. Calculus of Variations
Barbara Wendelberger
Logan Zoellner
Matthew Lucia

2. Motivation
Dirichlet Principle – One stationary ground state for energy
Solutions to many physical problems require maximizing or minimizing some parameter I.
Distance
Time
Surface Area
Parameter I dependent on selected path u and domain of interest D:
Terminology:
Functional – The parameter I to be maximized or minimized
Extremal – The solution path u that maximizes or minimizes I

3. Analogy to Calculus Single variable calculus:
Functions take extreme values on bounded domain.
Necessary condition for extremum at x0, if f is differentiable:
Calculus of variations:
True extremal of functional for unique solution u(x)
Test function v(x), which vanishes at endpoints, used to find extremal:
Necessary condition for extremal:

4. Solving for the Extremal
Differentiate I[e]:
Set I[0] = 0 for the extremal, substituting terms for e = 0 :
Integrate second integral by parts:

5. The Euler-Lagrange Equation
Since v(x) is an arbitrary function, the only way for the integral to be zero is for the other factor of the integrand to be zero. (Vanishing Theorem)
This result is known as the Euler-Lagrange Equation
E-L equation allows generalization of solution extremals to all variational problems.

6. Functions of Two Variables
Analogy to multivariable calculus:
Functions still take extreme values on bounded domain.
Necessary condition for extremum at x0, if f is differentiable:
Calculus of variations method similar:

7. Further Extension
With this method, the E-L equation can be extended to N variables:
In physics, the q are sometimes referred to as generalized position coordinates, while the uq are referred to as generalized momentum.
This parallels their roles as position and momentum variables when solving problems in Lagrangian mechanics formulism.

8. Limitations
Method gives extremals, but doesn’t indicate maximum or minimum
Distinguishing mathematically between max/min is more difficult
Usually have to use geometry of physical setup
Solution curve u must have continuous second-order derivatives
Requirement from integration by parts
We are finding stationary states, which vary only in space, not in time
Very few cases in which systems varying in time can be solved
Even problems involving time (e.g. brachistochrones) don’t change in time

20. Calculus of Variations
Examples in Physics Minimizing, Maximizing, and Finding Stationary Points (often dependant upon physical properties and geometry of problem)

21. Geodesics A locally length-minimizing curve on a surface
Find the equation y = y(x) of a curve joining points (x1, y1) and (x2, y2) in order to minimize the arc length
and so
Geodesics minimize path length

22. Fermat’s Principle
Refractive index of light in an inhomogeneous medium
, where v = velocity in the medium and n = refractive index
Time of travel =
Fermat’s principle states that the path must minimize the time of travel.

23. Brachistochrone Problem
Finding the shape of a wire joining two given points such that a bead will slide (frictionlessly) down due to gravity will result in finding the path that takes the shortest amount of time.
The shape of the wire will minimize time based on the most efficient use of kinetic and potential energy.

24. Principle of Least Action
Energy of a Vibrating String
Action = Kinetic Energy – Potential Energy
at ε = 0
Explicit differentiation of A(u+εv) with respect to ε
Integration by parts
v is arbitrary inside the boundary D
This is the wave equation!
Calculus of variations can locate saddle points
The action is stationary

25. Soap Film
When finding the shape of a soap bubble that spans a wire ring, the shape must minimize surface area, which varies proportional to the potential energy.
Z = f(x,y) where (x,y) lies over a plane region D
The surface area/volume ratio is minimized in order to minimize potential energy from cohesive forces.