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# a11f03 - L i = ± ijk x j p k = ~ r × ~ p i holds for i =...

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INTERMEDIATE MECHANICS II — PHY 4936 HOME AND CLASS WORK – SET 3 (September 16, 2011) (10) Verify the solution of Problem 3 (a), p.11 of Landau-Lifschitz along the following steps: Calculate as function of the generalized coordinate φ 10.1 ˙ x and ˙ y . 10.2 ˙ x 2 and ˙ y 2 . 10.3 Eliminate the term with ˙ φ sin( φ - γt ) from L . Due Friday, September 16 before class (10 points). (11a) Calculate explicitly δ x L for the Lagrangian of two harmonically bound particles in 1D L = m 1 2 ( ˙ x 1 ) 2 + m 2 2 ( ˙ x 2 ) 2 - k ( x 1 - x 2 ) 2 . Is there a conservation law? Due in class (2 points). (11b) Repeat the previous excercise for the harmonic ocillator in 1D L = m 1 2 ( ˙ x ) 2 - k 2 x 2 . Due in class (2 points). (12a) Use the Einstein summation convention in the following. Show that
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Unformatted text preview: L i = ± ijk x j p k = [ ~ r × ~ p ] i holds for i = 1 , 2 , 3 . Due in class (2 points). (12b) Express ± 12 k ± lmk in terms of Kronecker delta and then ± ijk ± lmk . Due in class (2 points). (13) Derive the answers of problems 1 and 2, Landau-Lifschitz, p.21. Due Monday, September 26 before class (10 points). (14) Derive the solutions of problems 1 and 2, Landau-Lifshitz p.24 Due Friday, September 30 (6 points). (15) Compute the solution for the equation of motion m ¨ x = k x 3 , k > with t = 0 initial values x > 0 and ˙ x = 0. Due Monday, October 3 before class (10 points)....
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