Calculus of Variations. Barbara Wendelberger Logan Zoellner Matthew Lucia. Motivation. Dirichlet Principle – One stationary ground state for energy Solutions to many physical problems require maximizing or minimizing some parameter I . Distance Time Surface Area
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Examples in PhysicsMinimizing, Maximizing, and Finding Stationary Points(often dependant upon physical properties and geometry of problem)
A locally length-minimizing curve on a surface
Find the equation y = y(x) of a curve joining points (x1, y1) and (x2, y2) in order to minimize the arc length
Geodesics minimize path length
Refractive index of light in an inhomogeneous medium
, where v = velocity in the medium and n = refractive index
Time of travel =
Fermat’s principle states that the path must minimize the time of travel.
Finding the shape of a wire joining two given points such that a bead will slide (frictionlessly) down due to gravity will result in finding the path that takes the shortest amount of time.
The shape of the wire will minimize time based on the most efficient use of kinetic and potential energy.
Energy of a Vibrating String
Action = Kinetic Energy – Potential Energy
at ε = 0
Explicit differentiation of A(u+εv) with respect to ε
Integration by parts
v is arbitrary inside the boundary D
This is the wave equation!
When finding the shape of a soap bubble that spans a wire ring, the shape must minimize surface area, which varies proportional to the potential energy.
Z = f(x,y) where (x,y) lies over a plane region D
The surface area/volume ratio is minimized in order to minimize potential energy from cohesive forces.