Variational Calculus

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

Path Variation  A trajectory y in space is a parametric function. y ( , x ) = y (0, x ) +  ( x )y ( , x ) = y (0, x ) +  ( x ) Continuous variation  ( x )Continuous variation  ( x ) End points  ( x 1 ) =  ( x 2 ) = 0End points  ( x 1 ) =  ( x 2 ) = 0  Define a function f in space.  Minimize the integral J. If y is varied J must increaseIf y is varied J must increase x2x2 x1x1 y(x)y(x) y( , x)

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

Boundary Conditions  The second term can be evaluated with integration by parts. Fixed at boundaries  (x 1 ) =  (x 2 ) = 0

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

Soap Film y (x 2, y 2 ) (x 1, y 1 ) 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.

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

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