Calculus of Variations

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Calculus of Variations
Barbara Wendelberger Logan Zoellner Matthew Lucia
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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
• Parameter I dependent on selected path u and
  domain of interest D
• Terminology
• Functional The parameter I to be maximized or
  minimized
• Extremal The solution path u that maximizes or
  minimizes I

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Analogy to Calculus

• Single variable calculus
• Functions take extreme values on bounded domain.
• Necessary condition for extremum at x0, if f is
  differentiable

• Calculus of variations
• True extremal of functional for unique solution
  u(x)
• Test function v(x), which vanishes at endpoints,
  used to find extremal
• Necessary condition for extremal

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Solving for the Extremal

• Differentiate Ie
• Set I0 0 for the extremal, substituting
  terms for e 0
• Integrate second integral by parts

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The Euler-Lagrange Equation

• Since v(x) is an arbitrary function, the only
  way for the integral to be zero is for the other
  factor of the integrand to be zero. (Vanishing
  Theorem)
• This result is known as the Euler-Lagrange
  Equation
• E-L equation allows generalization of solution
  extremals to all variational problems.

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Functions of Two Variables

• Analogy to multivariable calculus
• Functions still take extreme values on bounded
  domain.
• Necessary condition for extremum at x0, if f is
  differentiable

• Calculus of variations method similar

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Further Extension

• With this method, the E-L equation can be
  extended to N variables
• In physics, the q are sometimes referred to as
  generalized position coordinates, while the uq
  are referred to as generalized momentum.
• This parallels their roles as position and
  momentum variables when solving problems in
  Lagrangian mechanics formulism.

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Limitations

• Method gives extremals, but doesnt indicate
  maximum or minimum
• Distinguishing mathematically between max/min is
  more difficult
• Usually have to use geometry of physical setup
• Solution curve u must have continuous
  second-order derivatives
• Requirement from integration by parts
• We are finding stationary states, which vary
  only in space, not in time
• Very few cases in which systems varying in time
  can be solved
• Even problems involving time (e.g.
  brachistochrones) dont change in time

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Examples in PhysicsMinimizing, Maximizing, and
Finding Stationary Points(often dependant upon
physical properties and geometry of problem)
Calculus of Variations
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Geodesics

• 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
and so
Geodesics minimize path length
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Fermats Principle

• Refractive index of light in an inhomogeneous
  medium

, where v velocity in the medium and
n refractive index Time of travel
Fermats principle states that the path must
minimize the time of travel.
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Brachistochrone Problem
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.
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Principle of Least Action
Energy of a Vibrating String
Action Kinetic Energy Potential Energy
at e 0 Explicit differentiation
of A(uev) with respect to e Integration by
parts v is arbitrary inside the boundary D
This is the wave
equation!

• Calculus of variations can locate saddle points
• The action is stationary

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Soap Film
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.