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Euler's Equation

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					Euler’s Equation

Find the Extremum


Define a function along a trajectory.
• • • • y(a,x) = y(0,x) + ah(x) Parametric function Variation h(x) is C1 function. End points h(x1) = h(x2) = 0 x2

y(a, x) y(x)



Find the integral J
• If y is varied J must increase

x1

J   f ( y( x), y' ( x); x)dx
x1

x2

Parametrized Integral


Write the integral in parametrized form. Condition for extremum

J   f ( y(a , x), y' (a , x); x)dx
x1

x2



J a

0
a 0

for all h(x)



Expand with the chain rule
• Term a only appears with h

x2 f y J f y  (  )dx x1 y a a y a x2 f J f dh   ( h ( x)  )dx x1 y a y dx



Apply integration by parts …

Euler’s Equation
 
x2 x1 x2

(

x2 d f dh f f x )dx  h ( x ) x2   ( )h ( x)dx 1 x1 dx y  y dx y

h(x1) = h(x1) = 0

x1

x2 d f dh f ( )dx    ( )h ( x)dx x1 dx y  y dx

x2  f x2  f  J d  f  d  f     h ( x)    y h ( x)dx  x1  y  dx  y h ( x)dx    x1 a dx      y  

f d  f    0 y dx  y   

It must vanish for all h(x)
This is Euler’s equation

Geodesic




A straight line is the shortest distance between two points in Euclidean space. Curves of minimum length are geodesics.
• Tangents remain tangent as they move on the geodesic • Example: great circles on the sphere



Euler’s equation can find the minimum path.

Soap Film


Find a surface of revolution.
• Find the area • Minimize the function y (x2, y2)

f  x 1  y 2
f d  f  d   xy   0    y   0  dx   y dx  1  y 2     xy a 2 1  y
y  a x a
2 2

(x1, y1)

dA  2 dx 2  dy 2
x2 x1





 y b  x  a cosh  a  

A  2  x 1  y2 dx

Action


Motion involves a trajectory in configuration space Q.
• Tangent space TQ for full description.



The integral of the Lagrangian is the action.
 S   L(q j , q j ; t )dt
t1 t2



Find the extremum of action
• Euler’s equation can be applied to the action • Euler-Lagrange equations

Q q’

q

L d  L    j 0 j q dt  q    

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