Calculus of Variations Barbara Wendelberger Logan Zoellner Matthew Lucia Motivat
Halladay, Tom, Host has reference to this Academic Journal, PHwiki organized this Journal Calculus of Variations Barbara Wendelberger Logan Zoellner Matthew Lucia Motivation Dirichlet Principle One stationary ground state as long as energy Solutions to many physical problems require maximizing or minimizing some parameter I. Distance Time Surface Area Parameter I dependent on selected path u in addition to domain of interest D: Terminology: Functional The parameter I to be maximized or minimized Extremal The solution path u that maximizes or minimizes I Analogy to Calculus Single variable calculus: Functions take extreme values on bounded domain. Necessary condition as long as extremum at x0, if f is differentiable: Calculus of variations: True extremal of functional as long as unique solution u(x) Test function v(x), which vanishes at endpoints, used to find extremal: Necessary condition as long as extremal:
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Solving as long as the Extremal Differentiate I[e]: Set I = 0 as long as the extremal, substituting terms as long as e = 0 : Integrate second integral by parts: The Euler-Lagrange Equation Since v(x) is an arbitrary function, the only way as long as the integral to be zero is as long as the other factor of the integr in addition to 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. Functions of Two Variables Analogy to multivariable calculus: Functions still take extreme values on bounded domain. Necessary condition as long as extremum at x0, if f is differentiable: Calculus of variations method similar:
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 in addition to momentum variables when solving problems in Lagrangian mechanics as long as mulism. 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
Examples in Physics Minimizing, Maximizing, in addition to Finding Stationary Points (often dependant upon physical properties in addition to geometry of problem) Calculus of Variations Geodesics A locally length-minimizing curve on a surface Find the equation y = y(x) of a curve joining points (x1, y1) in addition to (x2, y2) in order to minimize the arc length in addition to so Geodesics minimize path length
Fermats Principle Refractive index of light in an inhomogeneous medium , where v = velocity in the medium in addition to n = refractive index Time of travel = Fermats principle states that the path must minimize the time of travel. 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 in addition to potential energy. Principle of Least Action Calculus of variations can locate saddle points The action is stationary Energy of a Vibrating String Action = Kinetic Energy Potential Energy at = 0 Explicit differentiation of A(u+v) with respect to Integration by parts v is arbitrary inside the boundary D This is the wave equation!
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 as long as ces.
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