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Optimizing Paths with Euler's Equation: Geodesics and Variational Calculus

Discover how Euler's Equation is utilized to find extremum paths and geodesics in variational calculus, demonstrating the shortest distances between points. Explore the application of the Euler-Lagrange equations in minimizing the action integral for motion trajectories.

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Optimizing Paths with Euler's Equation: Geodesics and Variational Calculus

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  1. Euler’s Equation

  2. 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

  3. 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)

  4. Euler’s Equation h(x1) = h(x1) =0 It must vanish for all h(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. Find a surface of revolution. Find the area Minimize the function Soap Film y (x2, y2) (x1, y1)

  7. 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 Euler’s equation can be applied to the action Euler-Lagrange equations Action Q q’ q next

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