Calculus of Variations

Barbara WendelbergerLogan ZoellnerMatthew 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:

• Terminology:• Functional – The parameter I to be maximized or minimized• Extremal – The solution path u that maximizes or minimizes I

( ), , xD

I F x u u dx= ∫

Analogy to Calculus• Single variable calculus:

• Functions take extreme values on bounded domain.• Necessary condition for extremum at x0, if f is differentiable:

( )0 0f x =′

• 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:

( ), ,b

xa

I F x w w dxε = ∫( ) ( ) ( )w x u x v xε= +

0dIdε =

Solving for the Extremal• Differentiate I[ε]:

• Set I[0] = 0 for the extremal, substituting terms for ε = 0 :

• Integrate second integral by parts:

( ) ( ), ,b b

xx

xa a

wdI d F w FF x w w dx dxw wd d

ε ε εε ε ÷ ÷

∂∂ ∂ ∂= = +∂ ∂ ∂ ∂∫ ∫

( ) ( )w v xεε∂ =∂ ( ) ( )x

xw v xεε

∂ =∂( )0w v xε ÷

∂ =∂ ( )0x xw v xε

÷ ∂ =∂

( )0w u x ÷ =

( )0x xw u x ÷ =

0b

xxa

dI F Fv v dxu udε

÷ ÷ ÷

∂ ∂= +∂ ∂∫ 0b b

xxa a

F Fvdx v dxu u

∂ ∂+ =∂ ∂∫ ∫

bb b b

xx x x xa a aa

F F d F d Fv dx v vdx vdxu u u udx dx

÷ ÷ ÷ ÷

∂ ∂ ∂ ∂= − = −∂ ∂ ∂ ∂∫ ∫ ∫

0x

F

u

b b

a a

F dvdx vdxu dx

∂ ÷∂

∂ − =∂∫ ∫ 0x

F d Fu dx u

b

avdx

÷ ÷

∂ ∂−∂ ∂

=∫

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.

0x

F d Fu dx u

b

avdx

÷ ÷

∂ ∂−∂ ∂

=∫

x

F d F

u dx u

∂ ∂= ∂ ∂

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:

( ) ( )0 0 0 0, , 0x yf x y f x y= =

• Calculus of variations method similar:

( ), , , ,x y

D

I F x y u u u dxdy= ∫∫ ( ) ( ) ( ), , ,w x y u x y v x yε= +

( ) ( ), , , , yxx y

x yD D

wwdI d F w F FF x y w w w dxdy dxdy

d d w w wε

ε ε ε ε ε ∂∂∂ ∂ ∂ ∂= = + + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂

∫∫ ∫∫

0x yx yD D D

F F Fvdxdy v dxdy v dxdy

u u u

∂ ∂ ∂+ + =∂ ∂ ∂∫∫ ∫∫ ∫∫

0x yD

F d F d Fvdxdy

u dx u dy u

∂ ∂ ∂− − = ÷ ÷ ÷∂ ∂ ∂ ∫∫ x y

F d F d F

u dx u dy u

∂ ∂ ∂= + ∂ ∂ ∂

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.

1i

N

i i q

F d F

u dq u=

∂ ∂= ∂ ∂

∑

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

Examples in Physics

Minimizing, Maximizing, and Finding Stationary Points(often dependant upon physical properties and

geometry of problem)

Calculus of Variations

GeodesicsA 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

2 2ds dx dy= + ( )dydy dx y x dx

dx′= =

( )

( )

2

2

1

1C C

ds y x dx

L ds y x dx

′= +

′= = +∫ ∫

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.

cv n=

( ) ( ) 2

1

, 1

C C C

C

dsT dt nds

v c

T n x y y x dx

= = =

′= +

∫ ∫ ∫

∫

Brachistochrone ProblemFinding 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.

( )

( ) ( )

2

2

11

11

,C C

dsv

dtds

dt y x dxv v

T dt y x dxv x y

=

′= = +

′= = +∫ ∫

Principle of Least Action

• Calculus of variations can locate saddle points

• The action is stationary

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!

[ ]2 2

D

u uA u T dxdt

t xρ

∂ ∂ = − ÷ ÷∂ ∂ ∫∫

( )dA u v

dε

ε+

[ ] 0D

u v u vA u T dxdt

t t x xρ ∂ ∂ ∂ ∂ = − = ÷ ÷ ÷ ÷ ∂ ∂ ∂ ∂ ∫∫

2 2

2 20

u uT

t xρ ∂ ∂− =

∂ ∂

[ ] [ ]2 2

2 20

2 2D

u T uA u v dxdt

t x

ρ ∂ ∂= − = ∂ ∂ ∫∫

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.

( ) ( ) ( ){ }2 2

, ;

1 x y

D

x y bdy D z h x

A u u dxdy

∈ =

= + +∫∫