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Unformatted text preview: 2 F âˆ‚q i âˆ‚q j Â· q i + âˆ‚ 2 F âˆ‚tâˆ‚q j . (7) Because it is immaterial in which order partial di f erentiation is carried out, we can rewrite Eq.(7) as d dt âˆ‚ Â· F âˆ‚ Â· q j = âˆ‚ âˆ‚q j " n X i =1 âˆ‚F âˆ‚q i Â· q i + âˆ‚F âˆ‚t # = âˆ‚ Â· F âˆ‚q j , (8) 1 using Eq.(5) to replace the term in square brackets. From this result, we see that Eq.(4) is indeed satis f ed and that, therefore, Eq.(3) is also valid. Thus, Lagrangeâ€™s EOM are invariant under the operation of adding to the Lagrangian the total time derivative of a function F that depends only on the generalized coordinates and time. A more general result of this sort will be useful to us in Chapters 9 and 10. 2...
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 Fall '10
 Wilemski
 mechanics, Derivative, dt âˆ‚ qj, Lagrange EOM

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