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1. Top School in Delhi NCRTop School in Delhi NCR
By:By:
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2. Euler’s EquationEuler’s Equation
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3. Find the ExtremumFind the Extremum
 Define a function along aDefine a function along a
trajectory.trajectory.
• yy((αα,,xx) =) = yy(0,(0,xx) +) + αηαη((xx))
• Parametric functionParametric function
• VariationVariation ηη((xx) is C) is C11
function.function.
• End pointsEnd points ηη((xx11) =) = ηη((xx22) = 0) = 0
 Find the integralFind the integral JJ
• IfIf yy is variedis varied JJ must increasemust increase
x2
x1
y(x)
y(α, x)
∫=
2
1
));('),((
x
x
dxxxyxyfJ
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4. Parametrized IntegralParametrized Integral
 Write the integral inWrite the integral in
parametrized form.parametrized form.
 Condition for extremumCondition for extremum
 Expand with the chain ruleExpand with the chain rule
• TermTerm αα only appears withonly appears with ηη
 Apply integration by parts …Apply integration by parts …
∫=
2
1
));,('),,((
x
x
dxxxyxyfJ αα
0
0
=
∂
∂
=αα
J
for all η(x)
dx
y
y
fy
y
fJ x
x
)(
2
1 ααα ∂
′∂
′∂
∂
+
∂
∂
∂
∂
=
∂
∂
∫
dx
dx
d
y
f
x
y
fJ x
x
))((
2
1
η
η
α ′∂
∂
+
∂
∂
=
∂
∂
∫
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5. Euler’s EquationEuler’s Equation
dxx
y
f
dx
d
x
y
f
dx
dx
d
y
f x
x
x
x
x
x
)()()()(
2
1
2
1
2
1
ηη
η
′∂
∂
−
′∂
∂
=
′∂
∂
∫∫ η(x1) = η(x1) = 0
dxx
y
f
dx
d
dx
dx
d
y
f x
x
x
x
)()()(
2
1
2
1
η
η
′∂
∂
−=
′∂
∂
∫∫
dxx
y
f
dx
d
y
f
dxx
y
f
dx
d
x
y
fJ x
x
x
x
)()()(
2
1
2
1
ηηη
α ∫∫ 











′∂
∂
−
∂
∂
=











′∂
∂
−
∂
∂
=
∂
∂
0=





′∂
∂
−
∂
∂
y
f
dx
d
y
f It must vanish for all η(x)
This is Euler’s equation
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6. GeodesicGeodesic
 A straight line is the shortestA straight line is the shortest
distance between two pointsdistance between two points
in Euclidean space.in Euclidean space.
 Curves of minimum lengthCurves of minimum length
areare geodesicsgeodesics..
• Tangents remain tangent asTangents remain tangent as
they move on the geodesicthey move on the geodesic
• Example: great circles onExample: great circles on
the spherethe sphere
 Euler’s equation can find theEuler’s equation can find the
minimum path.minimum path.
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7. Soap FilmSoap Film
 Find a surface of revolution.Find a surface of revolution.
• Find the areaFind the area
• Minimize the functionMinimize the function
y
( )22
2 dydxdA += π
dxyxA
x
x∫ ′+=
2
1
2
12π
2
1 yxf ′+=
0
1
0
2
=








′+
′−
−=





′∂
∂
−
∂
∂
y
yx
dx
d
y
f
dx
d
y
f
a
y
yx
=
′+
′
2
1
(x2, y2)
(x1, y1)
22
ax
a
y
−
=′





 −
=
a
by
ax coshSchool.edhole.com

8. ActionAction
 Motion involves a trajectoryMotion involves a trajectory
in configuration spacein configuration space QQ..
• Tangent spaceTangent space TTQQ for fullfor full
description.description.
 The integral of theThe integral of the
Lagrangian is theLagrangian is the actionaction..
 Find the extremum of actionFind the extremum of action
• Euler’s equation can beEuler’s equation can be
applied to the actionapplied to the action
• Euler-Lagrange equationsEuler-Lagrange equations
Q
q
q’
∫=
2
1
);,(
t
t
jj
dttqqLS 
0=





∂
∂
−
∂
∂
jj
q
L
dt
d
q
L

next
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