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na_lec_6a - The Euler-Lagrange Equation in Expanded Form F...

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The Euler-Lagrange Equation in Expanded Form 0 = y F dx d y F (1) 0 2 2 2 = dx y d y y F dx dy y y F dx dx y x F y F (2) 0 2 2 2 2 2 2 = dx y d y F dx dy y y F y x F y F (3) Special Case I: ( ) ' , y y F F = If ( ) ' , y y F F = . (4) Then the Euler-Lagrange Equation reduces to C y F y F = (5) where C is an arbitrary constant. Proof: ( ) dx y d y F y y F y y F dx d + = , (6) + = ' ' ' y F dx d y y F dx y d y F y dx d (7) 0 ' ' =
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