na_lec_6

na_lec_6 - Calculus of Variation 1. The Mathematical Tool -...

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Calculus of Variation 1. The Mathematical Tool - Taylor Expansion One-dimensional ( 29 ( 29 ( 29 ( 29 ( 29 ... ! 3 ! 2 ! 1 3 2 + ′′′ + ′′ + + = + x f x f x f x f x f e e e e Multiple-dimensional ( 29 ( 29 ( 29 ( 29 ( 29 .. ,..., , ! 3 ,..., , ! 2 ,..., , ! 1 ,..., , ,..., , 1 1 1 2 1 3 1 1 2 1 2 1 2 1 2 1 2 2 1 1 + + + + = + + + ∑∑∑ ∑∑ = = = = = = n i n j n k k j i n k j i n i n j j i n j i n i i n i n n n x x x x x x f x x x x x f x x x x f x x x f x x x f e e e e e e e e e 2. Motivation Problem - The Brachistochrone To find the shape of the curve connecting the two points ( 29 1 , y a A and ( 29 2 , y b B such that a particle starting its motion from rest at A will arrive, under the influence of gravity, at point B in the shortest time. Friction is neglected. Let us look on the infinitesimal interval of the path ds located at the position y x , . See the figure above. The velocity of the particle at this interval is ( 29 y y g v - = 1 2 . ( 29 x y ( 29 2 , y b B ( 29 1 , y a A ds ds dx dy x y
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Since dt ds v = we have ( 29 y y g ds v ds dt - = = 1 2 . Also dx dy y =′ or dx y dy = . We therefore have 2 2 2 ' 1 y dx dy dx ds + = + = and hence ( 29 dx y y g y dt - + = 1 2 2 ' 1 . The mathematical problem of the Brachistochrone is therefore to determine the function ( 29 x y that passes through A and B , and which minimizes the time integral ( 29 - + = b a t t dx y y g y dt 1 2 2 ' 1 2 1 .
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3. The Basic Problem of the Calculus of Variations The Brachistochrone problem leads to the investigation of the following problem. Given
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This note was uploaded on 12/02/2011 for the course ME 7533 taught by Professor Staff during the Summer '11 term at LSU.

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na_lec_6 - Calculus of Variation 1. The Mathematical Tool -...

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