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Euler’s EquationEuler’s Equation

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Find the ExtremumFind the Extremum

Define a function along a Define a function along a trajectory.trajectory.• yy((,,xx) = ) = yy(0,(0,xx) + ) + ((xx))

• Parametric functionParametric function

• Variation Variation ((xx) is C) is C11 function. function.

• End points End points ((xx11) = ) = ((xx22) = 0) = 0

Find the integral Find the integral JJ• If If yy is varied is varied JJ must increase must increase

x2

x1

y(x)

y(, x)

2

1

));('),((x

xdxxxyxyfJ

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Parametrized IntegralParametrized Integral

Write the integral in Write the integral in parametrized form.parametrized form.

Condition for extremumCondition for extremum

Expand with the chain ruleExpand with the chain rule• Term Term only appears with only appears with

Apply integration by parts …Apply integration by parts …

2

1

));,('),,((x

xdxxxyxyfJ

00

J

for all (x)

dxy

y

fy

y

fJ x

x)(

2

1

dxdx

d

y

fx

y

fJ x

x))((

2

1

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Euler’s EquationEuler’s Equation

dxxy

f

dx

dx

y

fdx

dx

d

y

f x

x

x

x

x

x)()()()(

2

1

2

1

2

1

(x1) = (x1) = 0

dxxy

f

dx

ddx

dx

d

y

f x

x

x

x)()()(

2

1

2

1

dxxy

f

dx

d

y

fdxx

y

f

dx

dx

y

fJ x

x

x

x)()()(

2

1

2

1

0

y

f

dx

d

y

f It must vanish for all (x)

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GeodesicGeodesic

A straight line is the shortest A straight line is the shortest distance between two points distance between two points in Euclidean space.in Euclidean space.

Curves of minimum length Curves of minimum length are are geodesicsgeodesics..• Tangents remain tangent as Tangents remain tangent as

they move on the geodesicthey move on the geodesic

• Example: great circles on Example: great circles on the spherethe sphere

Euler’s equation can find the Euler’s equation can find the minimum path.minimum path.

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Soap FilmSoap Film

Find a surface of revolution.Find a surface of revolution.• Find the areaFind the area

• Minimize the functionMinimize the function

y

222 dydxdA

dxyxAx

x 2

1

212

21 yxf

01

02

y

yx

dx

d

y

f

dx

d

y

f

ay

yx

21

(x2, y2)

(x1, y1)

22 ax

ay

a

byax coshSchool.edhole.com

ActionAction

Motion involves a trajectory Motion involves a trajectory in configuration space in configuration space QQ..• Tangent space Tangent space TTQQ for full for full

description.description.

The integral of the The integral of the Lagrangian is the Lagrangian is the actionaction..

Find the extremum of actionFind the extremum of action• Euler’s equation can be Euler’s equation can be

applied to the actionapplied to the action

• Euler-Lagrange equationsEuler-Lagrange equations

Q

qq’

2

1

);,(t

t

jj dttqqLS

0

jj q

L

dt

d

q

L

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