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Calculus of variations

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Calculus of variations

  1. 1. Calculus of Variations Barbara Wendelberger Logan Zoellner Matthew Lucia
  2. 2. 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: I = ò F x u u dx D • Terminology: ( , , x ) • Functional – The parameter I to be maximized or minimized • Extremal – The solution path u that maximizes or minimizes I
  3. 3. Analogy to Calculus • Single variable calculus: • Functions take extreme values on bounded domain. • Necessary condition for extremum at x0, if f is differentiable: f ¢( x0 ) = 0 • Calculus of variations: • True extremal of functional for unique solution u(x) • Test function v(x), which vanishes at endpoints, used to find extremal: ( , , ) b w( x) =u( x) +e v( x) I ée ù = F x w w dx ë û ò x a • Necessary condition for extremal: dI 0 de =
  4. 4. Solving for the Extremal • Differentiate I[e]: b b dI de ( e ) = d de ò F ( x , w , w ) dx = ò ¶ F ¶ w ¶ F ¶ w ¶ w ¶ e + ¶ w ¶ e dx æ ö ç ¸ ç ¸ è ø x x a a x • Set I[0] = 0 for the extremal, substituting terms for e = 0 : ew(e ) v( x) ¶ = 0 ( ) ¶ w v x e ¶ = ¶ ( ) ( ) x x w e v x e ¶ æç ö¸ = x ¶ 0 x ( ) è ø dI F v F v dx de u u = æ ¶ + ¶ ö ò ¶ ¶ 0 æç ö¸ ç ¸ è ø ç ¸ • Integrate second integral by parts: w v x e ¶ ¶ æç ö¸ = è ø wæçè0öø¸ u( x) = wx æçè0öø¸ ux ( x) = 0 b x a x è ø b b F vdx F v dx u u ò ¶ ¶ ¶ + ò¶ x = a a x b b b b F v dx F v d F vdx d F vdx u u dx u dx u ò¶ ¶ = ¶ - ò ¶ = - ò ¶ x ¶ ¶ ¶ é ù æ ö æ ö ê ú ç ¸ ç ¸ ê ú ç ¸ ç ¸ ë û è ø è ø a x x a a x a x ò ¶ ¶ - ò F = 0 0 u x b b a a F vdx d æ ¶ ö u dx ç ¸ vdx è ¶ ø F d F u dx u ¶ - ¶ ¶ ¶ ò = x b a vdx é æ öù ê çç ¸¸ú êë è øúû
  5. 5. The Euler-Lagrange • Since v(x) is an arbitraryE funqctionu, thae ontlyi woay fnor the integral to be zero is for the other factor of the integrand to be zero. (Vanishing Theorem) 0 vdx é æ öù F d F u dx u ¶ - ¶ ¶ ¶ ò = ê ¸¸ú êë çç è x øúû b a • This result is known as the Euler-Lagrange Equation ¶ F = d é ¶ F ù ¶ u dx ë¶ ê u ú x û • E-L equation allows generalization of solution extremals to all variational problems.
  6. 6. 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: ( 0, 0 ) ( 0, 0 ) 0 x y f x y = f x y = • Calculus of variations method similar: ( , , , , ) x y I = òò F x y u u u dxdy w( x, y) = u ( x, y) +e v ( x, y) D ( ) ( , , , , ) x y æ ¶ ¶ ¶ ¶ ¶ ¶ ö = = çç + + ¸¸ è ¶ ¶ ¶ ¶ ¶ ¶ ø dI d òò w e F x y w w w dxdy òò F w F w F dxdy d e d e x y w e w e w e D D x y F vdxdy F v dxdy F v dxdy u u u ¶ + ¶ + ¶ = òò ¶ òò ¶ òò ¶ 0 x y D D x D y ¶ é ¶ ù é ¶ ù = ê ú + ê ú ¶ ë¶ û êë¶ úû òò 0 x y é¶ F æ - d ¶ F ö d æ ¶ F öù ê ç ¸- çç ¸¸ú vdxdy = êë ¶ u dx è ¶ u ø dy è ¶ u øúû D x y F d F d F u dx u dy u
  7. 7. Further Extension • With this method, the E-L equation can be extended to N variables: ¶ F N é = å d ¶ F ù ê ú ¶ u i = 1 dq u i êë¶ q úû i • 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.
  8. 8. Limitations • Method gives extremals, but doesn’t 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) don’t change in time
  9. 9. Calculus of Variations Examples in Physics Minimizing, Maximizing, and Finding Stationary Points (often dependant upon physical properties and geometry of problem)
  10. 10. 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 ds = dx2 + dy2 dy and dy dx y ( x) dx so = = ¢ dx ( ) = + ¢ = ò = ò + ¢ ds y x dx L ds y x dx ( ) 2 2 Geodesics minimize path length 1 1 C C
  11. 11. Fermat’s Principle Refractive index of light in an inhomogeneous medium , where v = velocity in the medium and n = refractive index Time of travel = v c = n T dt ds 1 nds ò ò ò ò = = = v c C C C ( ) ( ) 2 = + ¢ T n x , y 1 y x dx C Fermat’s principle states that the path must minimize the time of travel.
  12. 12. 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. dt s y x dx ( ) 2 ( ) ( ) 2 1 1 v ds dtd v v 1 1 , C C T dt y x dx v x y = = = + ¢ = ò = ò + ¢
  13. 13. Principle of Least Action Energy of a Vibrating String • Calculus of variations can locate saddle points • The action is stationary Action = Kinetic Energy – Potential Energy é æ ¶ ö 2 æ ¶ ö 2 ù = ê ç ¸ - ú êë è ¶ ø è ç ¶ ø ¸ úû A u u T u dxdt t x at ε = 0 r d A( u + e v) d e Explicit differentiation of A(u+εv) with respect to ε A u u v T u v dxdt = éêr æç ¶ ö¸æç ¶ ö¸- æç ¶ ö¸æç ¶ ö¸ùú = ë è ¶ øè ¶ ø è ¶ øè ¶ øû òò [ ] 0 t t x x Integration by parts = ér ¶ - ¶ ù = ê ¶ ¶ ú ë û òò [ ] [ ] 2 2 A u u T u v dxdt 2 2 0 t x 2 2 D v is arbitrary inside the boundary D [ ] òò D D 2 2 2 2 u T u 0 t x r ¶ - ¶ = ¶ ¶ This is the wave equation!
  14. 14. 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. {( ) ( ) ( )} x , y bdy D ; z h x A u 2 u 2 dxdy 1 x y D Î = = òò + +

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