1. Euler’s Equation

2. Find the Extremum  Define a function along a trajectory. y ( , x ) = y (0, x ) +  ( x )y ( , x ) = y (0, x ) +  ( x ) Parametric functionParametric function Variation  ( x ) is C 1 function.Variation  ( x ) is C 1 function. End points  ( x 1 ) =  ( x 2 ) = 0End points  ( x 1 ) =  ( x 2 ) = 0  Find the integral J If y is varied J must increaseIf y is varied J must increase x2x2 x1x1 y(x)y(x) y( , x)

3. Parametrized Integral  Write the integral in parametrized form.  Condition for extremum  Expand with the chain rule Term  only appears with Term  only appears with   Apply integration by parts … for all  (x )

4. Euler’s Equation  (x 1 ) =  (x1) = 0 It must vanish for all  (x ) This is Euler’s equation

5. 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.

6. Soap Film  Find a surface of revolution. Find the areaFind the area Minimize the functionMinimize the function y (x 2, y 2 ) (x 1, y 1 )

7. Action  Motion involves a trajectory in configuration space Q. Tangent space T Q for full description.Tangent space T Q for full description.  The integral of the Lagrangian is the action.  Find the extremum of action Euler’s equation can be applied to the action Euler-Lagrange equations Q q q’ next