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.
for all h(x)
Fixed at boundaries
h(x1) =h(x2) = 0Boundary Conditions
The quantity in brackets must vanish.
This is Euler’s equation.
General mathematical relationshipEuler’s Equation
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