Calculus of Variations
Barbara Wendelberger
Logan Zoellner
Matthew Lucia
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:
I   F  x,u,ux  dx
D
• Terminology:
• Functional – The parameter I to
be maximized or minimized
• Extremal – The solution path u
that maximizes or minimizes I
Analogy to Calculus
• Single variable calculus:
• Functions take extreme values on
bounded domain.
• Necessary condition for extremum
at x0, if f is differentiable:
f   x0   0
• Calculus of variations:
• True extremal of functional for
unique solution u(x)
• Test function v(x), which vanishes
at endpoints, used to find extremal:
w x   u  x    v  x 
b
I    F  x, w, wx  dx
 
 
a
• Necessary condition for extremal:
dI
d
0
Solving for the Extremal
• Differentiate I[]:
dI   d F x, w, w dx   F w  F wx dx
x


d   d a 
a  w  wx  
b
b
• Set I[0] = 0 for the extremal, substituting terms for  = 0 :
w   v x
    
w  0   v x
 
    
dI  0    F
 
d   a  u
b
wx   v x wx  0   v x
 
   x      x  
F
F
a u vdx  a ux vxdx  0
b
b
• Integrate second integral by parts:
b
b
b
F v dx   F v   d  F  vdx   d  F  vdx
x


 ux 
dx  ux 
a ux
a dx  ux 

a a


F vdx  d  F  vdx  0

  u 
a u
a dx 
b
b
x
 
wx  0   ux  x
 

v  F vx dx
ux 
b
w 0   u  x 

b
F d  F  
 

 

u
dx

u

 x  
a

vdx  0
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)
b
F d  F  
 



u
dx  ux  


a

vdx  0
• This result is known as the Euler-Lagrange Equation
F d  F 
 

u dx  ux 
• E-L equation allows generalization of solution
extremals to all variational problems.
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:
f x  x0 , y0   f y  x0 , y0   0
• Calculus of variations method similar:
I   F  x, y, u, ux , u y  dxdy
w  x, y   u  x, y    v  x, y 
D
 F w F wx F wy 
D F  x, y, w, wx , wy  dxdy  D  w   wx   wy   dxdy


F
F
F
vdxdy

v
dxdy

D u
D ux x
D u y vy dxdy  0
dI
d
  
d
d
 F d  F  d  F
D  u  dx  ux   dy  u y



  vdxdy  0
 
F d  F  d  F 



 
u dx  u x  dy  u y 
Further Extension
• With this method, the E-L equation can be extended to N variables:
F N d

u i 1 dqi
 F

 uqi



• 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.
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
Calculus of Variations
Examples in Physics
Minimizing, Maximizing, and Finding Stationary Points
(often dependant upon physical properties and
geometry of problem)
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
ds  dx 2  dy 2
and
so
dy 
dy
dx  y  x  dx
dx
ds  1  y  x  dx
2
L   ds   1  y  x  dx
2
C
C
Geodesics minimize path length
Fermat’s Principle
Refractive index of light in an inhomogeneous
medium
vc
n , where v = velocity in the medium and n = refractive index
ds 1
Time of travel = T   dt  
  nds
v cC
C
C
T   n  x, y  1  y  x  dx
2
C
Fermat’s principle states that the path must minimize the time of travel.
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.
ds
dt
ds 1
2
dt 

1  y  x  dx
v v
1
2
T   dt  
1  y  x  dx
v x, y 
C
C 
v
The shape of the wire will minimize
time based on the most efficient
use of kinetic and potential energy.
Principle of Least Action
Energy of a Vibrating String
Action = Kinetic Energy – Potential Energy
• Calculus of
variations can
locate saddle points
• The action is
stationary
2
  u 2
 u  
A u         T    dxdt
 x  
  t 
D 
d
A u   v 
d
at ε = 0
Explicit differentiation of A(u+εv) with
respect to ε
  u  v 
 u  v  
A u          T     dxdt  0
t  t 
 x  x  
D  
Integration by parts
   2u T  2u 
Au    

v dxdt  0
2
2  
2

t
2

x

D 
v is arbitrary inside the boundary D
 2u
 2u
 2 T 2  0
t
x
This is the wave equation!
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
 x, y   bdy  D  ; z  h  x 
A   1  u x2  u y2 dxdy
D
The surface area/volume ratio is minimized
in order to minimize potential energy from
cohesive forces.