Euler’s Equation

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

x2
y(a, x)
y(x)
x1
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Parametrized Integral

• Write the integral in parametrized form.
• Condition for extremum
• Expand with the chain rule
• Term a only appears with h
• Apply integration by parts

for all h(x)
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Eulers Equation
h(x1) h(x1) 0
It must vanish for all h(x) This is Eulers
equation
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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
• Eulers equation can find the minimum path.

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Soap Film

• Find a surface of revolution.
• Find the area
• Minimize the function

y
(x2, y2)
(x1, y1)
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Action

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

• The integral of the Lagrangian is the action.
• Find the extremum of action
• Eulers equation can be applied to the action
• Euler-Lagrange equations

Q
q
q
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