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Lecture 10 Variational Calculus PH101 Lecture 10 Variational Calculus History of Variational Calculus (Wikipedia/Rana&Joag) Pierre de Fermat Isaac Newton Jacob Bernoulli (1607 –1665) (1642 – 1727) Johann (Jean or John) (1655 – 1705) Bernoulli ( 1667 –1748) Algebra Variational calculus Leonhard Euler Joseph-Louis Daniel Bernoulli ( 1700 –1782) (1707-1783) Lagrange Bernoulli's principle on fluids Fermat’s principle of least (extremum) time ~1662 speed of light in vacuum n = speed of light in medium www.physicsforums.com Jean Bernoulli’s challenge! “Brachistochrone” What should be the shape of a stone’s trajectory (or, of a roller coaster track) so that released from point 1 it reaches point 2 in the shortest possible time? Brachistochrone problem! ~1696 1 X Brachisto ~ shortest Chrone ~ time dx ͦ ͦ ͦ 4 ͦ ds dy ͧ͘ = ͬ͘ ƍ ͭ͘ = 1 ƍ 3 ͬ͘ = 1 ƍ ͭ′ ͬ͘ v= -Y 2 2͛ͭ v Time (from 1 to 2) I ͦ . = ͦ ͥͮ4ɑ = ͦ ɑ ¢ Ï, Ï , Î ºÎ = Ȅͥ 1 Ȅͥ ͦ"4 ͬ͘ Ȅͥ Cycloid Wiki Extremums! Can we prove mathematically Shortest distance between two points is a straight line? Shortest path between two points on the surface of a sphere is along the great-circle ? To answer these questions, one need to know necessary condition that the integral Î̋ ɑ where ºÏ ̏ = ȄÎ̊ ¢ Ï, Ï , Î ºÎ, Ï = Ï Î , Ï′ = ºÎ is stationary (ie, an extremum ! – either a maximum or a minimum !) . Interestingly we are already familiar with the solution ! Extremum of a function To locate the minimum, x0 ͙͑ ͕ͪͦͭ, Î → ͕͘͢ É»·È¹¾ ͚ͣͦ ͙ͨ͜ ͕ͣ͗ͨͣ͢͠͝= f(x+ x)- f(x) → 0 (for͚ small x) f(x+ x) f(x) x x 0 ͬ ƍ ͬ We say the function is stationary at, x0 (Meaning, for small steps, x, at x0 the value of the function does not change. = i! x = 0 ͚ i3 3ͤ Possible integration paths Out of the infinite number of possible paths, , which path makes the integral, ͭ(ͬ) Î̋ -stationary ? ̏ = ȄÎ̊ ¢ Ï, Ï′, Î ºÎ Y B A X ͬͥ ͬͦ We need to find the condition for an integral to be stationary, where the variable is a “function” itself [ ] -the integration path! Ï = Ï(Î) Smart choice of varied paths Step 1: Let’s assume as the path for which integral, ͭ(ͬ) = ͓ ͬ 3v is stationary . I = Ȅ3u ̀ ͭ, ͭʖ, ͬ ͬ͘ Step2: can represent all possible paths between and forͭ(ͬ) different = ͓ ͬ ƍ ∆ͭ(ͬ). ͬͥ ͬͦ ∆ͭ(ͬ) Can you choose suitable mathematical form of such that ∆Ï(Î) should represent all varied paths but must not (i) haveͭ(ͬ) variations = ͓ ͬ ƍ at ∆ͭ(ͬ) A( and (fixed points). ͬͥ) ̼(ͬͦ) goes to zero in the limiting case when the varied paths are very (ii) close∆ͭ(ͬ) to . ͓(ͬ) Let’s check this choice ∆Ï(Î) = ą ñ(Î) where is any arbitrary function of such that η(ͬ) . [condition (i) satisfied]ͬ η ͬͥ = η ͬͦ = 0 is a parameter which can vary from 0 to higher value# continuously. If we take limit , then condition (ii) satisfied . ą → ̉ and are indeed smart choice ñ(Î) # Typical choice of arbitrary function Y ͭ = ͓ ͬ ƍ 0.9(ͬ) , B ñ(Î) . η ͬͥ = η ͬͦ = 0 ͭ = ͓(ͬ) By varying , different strength of # (ͬ) ͭ = ͓ ͬ ƍ 0.05(ͬ) can be added to to generate range A of possible paths͓(ͬ) between A and B. ñ(Î) X gives us the true path # → 0 ͓ ͬ . ͬͥ ͬͦ For another choice of to generate Y ͭ = ͓ ͬ ƍ 1.2 (ͬ) another series of possibleη ͬ paths between A B and B by varying . # ͭ = ͓(ͬ) Thus arbitrary and can produce ͭ = ͓ ͬ ƍ 0.03(ͬ) ñ Î # A all possible paths. ñ(Î) X ͬͥ ͬͦ Stationary condition of integral Step 3 : Variation of the integral value for different paths nearby to the stationary path ], is negligibly small. ͓(ͬ) [͙͝, # → 0 ͙͙͗͜͢ ∆ͭ → 0 The meaning of the statement is Î̋ along stationary path IntegRation ̏ = Ȅ ¢ µ, µ′, Î ºÎ y(x) = ͓(ͬ) and Î̊ along the nearby paths [ ¿ÄÊ»½È·Ê¿ÅÄ Ï Î, ą = ͓ ƍ ∆ͭ = µ Î ƍ ąη Î, ·Äº ą → ̉] Î̋ Î̋ ̏(̃) = ǹ ¢ µ ƍ ∆Ï , µ′ ƍ ∆Ï′ , Î ºÎ] = ǹ ¢ Ï Î, ą , ͭ′ Î, ą , Î ºÎ Î̊ Î̊ must be equal . i.e, , ̓(̃) = 0 # → 0 This can be achieved by, º¥(ą) ʺ = ̉ ºą ą→̉ For stationary path º¥ (ą) ʺ = ̉ ºą ą→̉ 3v Where, ̓͘(#) ͘ ˄ = ǹ ̀ ͭ ͬ, # , ͭ′ ͬ,# ,ͬ ͬ͘ Ï Î, ą = µ ƍ ąη ͘# j→ͤ ͘# 3u ͭ′ Î, ą = µ′ ƍ ąη′ 3v 3v 3v "̀ "ͭ "̀ "ͭ′ "̀ "̀ = ǹ ƍ ͬ͘ = ǹ ͬ͘ ƍ ǹ ′ͬ͘ "ͭ "# "ͭ′ "# "ͭ "ͭ′ 3u 3u 3u 3v 3v 3ͦ Integration by parts "̀ "̀ ͘ "̀ = ǹ ͬ͘ ƍ ˄ − ǹ ͬ͘ Î̋ "ͭ "ͭ′ ͬ͘ "ͭ′ Ą¢ 3u 3u 3u =0 ±É¿Ä½ ĄÏɑ ηʺ 3v Î̊ ͘ "̀ "̀ As = − ǹ − ͬ͘ = 0 η ͬͥ = η ͬͦ = 0 ͬ͘ "ͭ′ "ͭ 3u This equation is true for any possible choice of , thus η(Î) º Ą¢ Ą¢ =0, Euler-Lagrange equation! ºÎ Ą4ɑ − ĄÏ This is necessary condition for 3v to be stationary! I = Ȅ3u ̀ ͭ, ͭʖ, ͬ ͬ͘ Application of Variational principle: Example1 Given two points in a plane, what is the shortest path between them? We certainly know the answer: Straight line. Let’s prove it using variation method B( ͬͦ, ͭͦ) ºÉ ºÏ ºÎ A( ͬͥ, ͭͥ) Consider an arbitrary path elementary length ͭ ͬ , u ͦ ⁄v ͦ ͦ 4 ͦ ͧ͘ = ͬ͘ ƍ ͭ͘ = 1 ƍ 3 ͬ͘ = 1 ƍ ͭ′ ͬ͘ Total path length 3v Ȅ ͧ͘ = Ȅ 1 ƍ ͭ′ͦ ͬ͘ Necessary condition for this 3integralu to be stationary (maximum) º Ą¢ Ą¢ =0; Here ɑ2 u ɑ Ɵv ºÎ ĄÏ − ĄÏ ̀ ͭ, ͭ′, ͬ = 1 ƍ ͭ Application of variational principle: Example1 ɑ "̀ " ɑ2 ͭ ′ = ′ 1 ƍ ͭ = "ͭ "ͭ 1 ƍ ͭɑ2 i 4ɒ = A const. ͦ i4 = 0 → ͥͮ4ɒ Thus ͭ′ͦ = ̻2 1 ƍ ͭ′ ͦ ͦ ͭɑ (1 − ̻2) = ̻2 ͦ ɑ ͦ ͭ = (ͥͯ ) = ͡ ͭ ͬ = ͬ͡ ƍ ̽, Equation of straight͙͙͑ͦ͜ ͡ ͕͘͢ ̽ ͕͙ͦ ͕͗ͣͧͨͨ͢͢ line. Shortest distance between two points in a plane is straight line . Questions?.
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