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Unformatted text preview: A Corollary of Eulerâ€™s Equations Shina Tan 1 The Corollary From Eulerâ€™s equations âˆ‚f âˆ‚y i d dx âˆ‚f âˆ‚y i = 0 , i = 1 , 2 , Â·Â·Â· , n, we can derive an important Corollary : d dx f n X i =1 y i âˆ‚f âˆ‚y i ! = âˆ‚f âˆ‚x . (1) In particular, if the function f ( y 1 ,y 1 ; y 2 ,y 2 ; Â·Â·Â· ; y n ,y n ; x ) does not explicitly depend on x , we have âˆ‚f âˆ‚x = 0 and f n X i =1 y i âˆ‚f âˆ‚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 âˆ‚f âˆ‚y i ! âˆ‚f âˆ‚x = df dx " n X i =1 d dx y i âˆ‚f âˆ‚y i # âˆ‚f âˆ‚x = (" n X i =1 âˆ‚f âˆ‚y i dy i dx + âˆ‚f âˆ‚y i dy i dx # + âˆ‚f âˆ‚x dx dx ) " n X i =1 dy i dx âˆ‚f âˆ‚y i + y i d dx âˆ‚f âˆ‚y i # âˆ‚f âˆ‚x = " n X i =1 y i âˆ‚f âˆ‚y i + n X i =1 y 00 i âˆ‚f âˆ‚y i # + âˆ‚f âˆ‚x " n X i =1 y 00 i âˆ‚f âˆ‚y i + n X i =1 y i d dx âˆ‚f âˆ‚y i # âˆ‚f âˆ‚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.
 Spring '11
 Tan
 mechanics

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