L-vp - A Corollary of Euler’s Equations Shina Tan 1 The Corollary From Euler’s equations ∂f ∂y i d dx ∂f ∂y i = 0 i = 1 2 ·· n we

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Unformatted text preview: A Corollary of Euler’s Equations Shina Tan 1 The Corollary From Euler’s equations ∂f ∂y i- d dx ∂f ∂y i = 0 , i = 1 , 2 , ··· , n, we can derive an important Corollary : d dx f- n X i =1 y i ∂f ∂y i ! = ∂f ∂x . (1) In particular, if the function f ( y 1 ,y 1 ; y 2 ,y 2 ; ··· ; y n ,y n ; x ) does not explicitly depend on x , we have ∂f ∂x = 0 and f- n X i =1 y i ∂f ∂y i = constant . (2) This result not only is useful in its own right (see next page), but also will shed a new light on the law of energy conservation when we get to Hamiltonian dynamics. 2 Proof d dx f- n X i =1 y i ∂f ∂y i !- ∂f ∂x = df dx- " n X i =1 d dx y i ∂f ∂y i #- ∂f ∂x = (" n X i =1 ∂f ∂y i dy i dx + ∂f ∂y i dy i dx # + ∂f ∂x dx dx )- " n X i =1 dy i dx ∂f ∂y i + y i d dx ∂f ∂y i #- ∂f ∂x = " n X i =1 y i ∂f ∂y i + n X i =1 y 00 i ∂f ∂y i # + ∂f ∂x- " n X i =1 y 00 i ∂f ∂y i + n X i =1 y i d dx ∂f ∂y i #- ∂f ∂x = n X i...
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This note was uploaded on 01/22/2012 for the course PHYS 3202 taught by Professor Tan during the Spring '11 term at Georgia Institute of Technology.

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L-vp - A Corollary of Euler’s Equations Shina Tan 1 The Corollary From Euler’s equations ∂f ∂y i d dx ∂f ∂y i = 0 i = 1 2 ·· n we

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