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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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Presentation Transcript
functional
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
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

integral extremum
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)

euler s equation
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
soap film
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)

euler applied
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
action
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

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