# L-vp - A Corollary of Eulers Equations Shina Tan 1 The...

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A Corollary of Euler’s Equations Shina Tan 1 The Corollary From Euler’s equations ∂f ∂y i - d dx ∂f ∂y 0 i = 0 , i = 1 , 2 , · · · , n, we can derive an important Corollary : d dx f - n X i =1 y 0 i ∂f ∂y 0 i ! = ∂f ∂x . (1) In particular, if the function f ( y 1 , y 0 1 ; y 2 , y 0 2 ; · · · ; y n , y 0 n ; x ) does not explicitly depend on x , we have ∂f ∂x = 0 and f - n X i =1 y 0 i ∂f ∂y 0 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 0 i ∂f ∂y 0 i ! - ∂f ∂x = df dx - " n X i =1 d dx y 0 i ∂f ∂y 0 i # - ∂f ∂x = (" n X i =1 ∂f ∂y i dy i dx + ∂f ∂y 0 i dy 0 i dx # + ∂f ∂x dx dx ) - " n X i =1 dy 0 i dx ∂f ∂y 0 i + y 0 i d dx ∂f ∂y 0 i # - ∂f ∂x = " n X i =1 y 0 i ∂f ∂y i + n X i =1 y 00 i ∂f ∂y 0 i # + ∂f ∂x - " n X i =1 y 00 i ∂f ∂y 0 i + n X i =1 y 0 i d dx ∂f ∂y 0 i # - ∂f ∂x = n X i =1 y 0 i ∂f ∂y i - n X i =1 y 0 i d dx ∂f ∂y 0 i = n X i =1 y 0 i ∂f ∂y i - d dx ∂f ∂y 0 i = 0

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