Calculus of Variations.1

7/31/2019 Calculus of Variations.1

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Calculus of Variations

Barbara WendelbergerLogan Zoellner

Matthew Lucia


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Motivation Dirichlet Principle One stationary

ground state for energy

Solutions to many physical problemsrequire maximizing or minimizing someparameter I.

Distance

Time Surface Area

Parameter Idependent on selectedpath uand domain of interest D:

Terminology: Functional The parameter Itobe maximized or minimized Extremal The solution path u

that maximizes or minimizes I

, ,

x

DI F x u u dx


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Analogy to Calculus Single variable calculus:

Functions take extreme values onbounded domain. Necessary condition for extremumat x0, if f is differentiable:

0 0f x

Calculus of variations: True extremal of functional forunique solution u(x)Test function v(x), which vanishesat endpoints, used to find extremal:

Necessary condition for extremal:

, ,b

xa

I F x w w dx

w x u x v x

0dI

d


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Solving for the Extremal Differentiate I[]:

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

Integrate second integral by parts:

, ,b b

xx

xa a

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

w

v x

x

x

w

v x

0w

v x

0

x

x

w

v x

0w u x

0x xw u x

0b

xxa

dI F F v v dxu ud

0

b b

xxa a

F Fvdx v dxu u

bb b b

xx x x xa a aa

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

0x

F

u

b b

a a

F dvdx vdxu dx

0

x

F d F

u dx u

b

a

vdx


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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 solutionextremals to all variational problems.

0x

F d F

u dx u

b

a

vdx

x

F d F

u dx u


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Functions of Two Variables Analogy to multivariable calculus:

Functions still take extremevalues on bounded domain. Necessary condition for extremumat x0, if f is differentiable:

0 0 0 0, , 0x yf x y f x y

Calculus of variations method similar:

, , , ,x yD

I F x y u u u dxdy , , ,w x y u x y v x y

, , , ,yx

x yx yD D

wwdI d F w F F

F x y w w w dxdy dxdyd d w w w

0

x y

x yD D D

F F Fvdxdy v dxdy v dxdy

u u u

0

x yD

F d F d F vdxdy

u dx u dy u

x y

F d F d F

u dx u dy u


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Further Extension With this method, the E-L equation can be extended to N variables:

In physics, the qare sometimes referred to as generalized positioncoordinates, while the uqare referred to as generalized momentum.

This parallels their roles as position and momentum variables when solving

problems in Lagrangian mechanics formulism.

1 i

N

i i q

F d F

u dq u


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LimitationsMethod gives extremals, but doesnt indicate maximum or minimum

Distinguishing mathematically between max/min is more difficult

Usually have to use geometry of physical setup

Solution curve umust 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) dont change in time


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Examples in Physics

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

geometry of problem)

Calculus of Variations


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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 orderto minimize the arc length

and

so

Geodesics minimize path length

2 2ds dx dy dy

dy dx y x dx

dx

2

2

1

1

C C

ds y x dx

L ds y x dx


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Fermats Principle

Refractive index of light in an inhomogeneousmedium

, where v= velocity in the medium and n= refractive index

Time of travel =

Fermats principle states that the path must minimize the time of travel.

cvn

2

1

, 1

C C C

C

dsT dt nds

v c

T n x y y x dx


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Brachistochrone Problem

Finding the shape of a wire joining two given points such thata bead will slide (frictionlessly) down due to gravity will resultin finding the path that takes the shortest amount of time.

The shape of the wire will minimizetime based on the most efficientuse of kinetic and potential energy.

2

2

11

11

,C C

dsv

dt

dsdt y x dx

v v

T dt y x dxv x y


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Principle of Least Action

Calculus ofvariations canlocate saddle points

The action isstationary

Energy of a Vibrating String

Action = Kinetic Energy Potential Energy

at = 0

Explicit differentiation of A(u+v) withrespect to

Integration by parts

vis arbitrary inside the boundary D

This is the wave equation!

2 2

D

u uA u T dxdt

t x

d

A u vd

0D

u v u vA u T dxdt

t t x x

2 2

2 20

u uT

t x

2 2

2 2 02 2D

u T u

A u v dxdtt x


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Soap Film

When finding the shape of a soap bubble that spans a wirering, the shape must minimize surface area, which variesproportional to the potential energy.

Z = f(x,y) where (x,y) lies over a plane region D

The surface area/volume ratio is minimizedin order to minimize potential energy fromcohesive forces.

2 2

, ;

1x y

D

x y bdy D z h x

A u u dxdy

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Calculus of Variations.1