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Variational Calculus

Variational Calculus. Functional. Calculus operates on functions of one or more variables. Example: derivative to find a minimum or maximum Some problems involve a functional. The function of a function Example: work defined on a path; path is a function in space. Path Variation.

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Variational Calculus

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  1. Variational Calculus

  2. Functional • Calculus operates on functions of one or more variables. • Example: derivative to find a minimum or maximum • Some problems involve a functional. • The function of a function • Example: work defined on a path; path is a function in space

  3. Path Variation • A trajectory y in space is a parametric function. • y(a,x) = y(0,x) + ah(x) • Continuous variation h(x) • End points h(x1) = h(x2) = 0 • Define a function f in space. • Minimize the integral J. • If y is varied J must increase y(a, x) y(x) x1 x2

  4. Integral Extremum • Write the integral in parametrized form. • May depend on y’ =dy/dx • Derivative on parameter a • Expand with the chain rule. • Term a only appears with h for all h(x)

  5. The second term can be evaluated with integration by parts. Fixed at boundaries h(x1) =h(x2) = 0 Boundary Conditions

  6. The variation h(x) can be factored out of the integrand. The quantity in brackets must vanish. Arbitrary variation This is Euler’s equation. General mathematical relationship Euler’s Equation

  7. Problem A soap film forms between two horizontal rings that share a common vertical axis. Find the curve that defines a film with the minimum surface area. Define a function y. The area A can be found as a surface of revolution. Soap Film y (x2, y2) (x1, y1)

  8. The area is a functional of the curve. Define functional Use Euler’s equation to find a differential equation. Zero derivative implies constant Select constant a The solution is a hyperbolic function. Euler Applied

  9. The time integral of the Lagrangian is the action. Action is a functional Extends to multiple coordinates The Euler-Lagrange equations are equivalent to finding the least time for the action. Multiple coordinates give multiple equations This is Hamilton’s principle. Action next

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