Euler's equation can be used to find curves or trajectories that extremize (minimize or maximize) functionals which are defined as integrals along curves. It does this by requiring the first variation of the integral, with respect to variations of the curve, to vanish. This results in an equation involving derivatives of the integrand which must be satisfied by any extremizing curve. Examples include finding geodesics as curves of shortest distance, surfaces of minimal area obtained by rotating curves, and trajectories that extremize the action in physics.

1. Admissions in India 2015Admissions in India 2015
By:By:
http://admission.edhole.comhttp://admission.edhole.com

2. Euler’s EquationEuler’s Equation

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

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 ′∂
∂
+
∂
∂
=
∂
∂
∫

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

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.

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 cosh

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