a11f06

# a11f06 - function of ˙ x 1 ˙ x 2 x 1 and x 2 Then...

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ADVANCED DYNAMICS — PHY 4936 HOME AND CLASS WORK – SET 6 (November 1, 2011) Read Landau-Lifshitz p.58 up to p.72 ( § 21 to § 24). (26) Continue with the double pendulum from assignment 25. 1. Use the eigenfrequencies ω ± given in the posted solution of 25 and normal coordinates (Landau-Lifshitz p.67/8) to write down the general solution for the two angles. 2. Express the integration constants of your solution through the angular positions and velocities at time t = 0, denoted by φ 0 , ˙ φ 0 , ψ 0 , ˙ ψ 0 . 3. Use q l/g as time unit and plot the solutions φ ( t ) and ψ ( t ) up to t = 50 q l/g for initial conditions φ 0 = 0 , ˙ φ 0 = q g/l, ψ 0 = 0 , ˙ ψ 0 = - q g/l . Due November 7 before class (10 points). (27) (A) Calculate the eigenfrequencies of a 2D harmonic oscillator ± 2 X k =1 m ik ¨ x k + k ik x k ! = 0 , ( i = 1 , 2) with matrix elements M = ( m ik ) = ² 1 1 1 2 ³ and K = ( k ik ) = ² 5 1 1 2 ³ . Due in class (4 points). (B) Use normal co-ordinates Θ 1 and Θ 2 as deﬁned in Landau-Lifshitz (p.67/68). Express x 1 and x 2 in terms of them. Due in class (4 points). (C) Use the given numbers for m ik and k ik and write down the Lagrangian as
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Unformatted text preview: function of ˙ x 1 , ˙ x 2 , x 1 and x 2 . Then, substitute normal co-ordinates as found in (B) and write down the Lagrangian in terms of ˙ Θ 1 , ˙ Θ 2 , Θ 1 and Θ 2 . Due November 7 before class (6 points). (D) Assume at time t = 0 the initial conditions Θ 1 (0) = 1, ˙ Θ 1 (0) = 0, Θ 2 (0) = 0 and ˙ Θ 2 (0) = 1. Plot the resulting solution ﬁrst in in the Θ 1-Θ 2 plane and then in the x 1-x 2 plane. Due November 9 before class (4 points). (28) The Lagrangian of a 2D oscillator is L = m 2 ´ ˙ x 2 + ˙ y 2-ω 2 1 x 2-ω 2 2 y 2 µ . Write down the general solution for the case that x = y = 0 at t = 0. Which condition applies to ω 1 and ω 2 , so that the mass point returns to x = y = 0 at some future time t ? How long will it take? Due November 14 before class (4 points). Read Landau-Lifshitz p.96 up to p.101 ( § 31 and § 32)....
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