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# a1004 - (18 Consider the spherical pendulum of mass m...

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ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 4 (January 27, 2010) (16) 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 February 3 in class (2 points). (17) Continue with the double pendulum from assignment 13. 1. Use eigenvalues ω ± as given in the posted solution and calculate corre- sponding eigenvectors in the form 1 ψ 0 ± . 2. Are these eigenvectors orthogonal? 3. Write down the general solution. 4. 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 . 5. 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 = 1 , ψ 0 = 0 , ˙ ψ 0 = - 1. Due February 1 before class (10 points).
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Unformatted text preview: (18) Consider the spherical pendulum of mass m again. (a) Use the Legendre transformation H ( θ, φ, p θ , p φ ) = ˙ θp θ + ˙ φp φ-L ( θ, φ, ˙ θ, ˙ φ ) , to construct the Hamiltonian of the system and show that it is identical to the energy E = T + V . (b) Write down Hamilton’s equations of motions for the system and identify a conserved quantity. Due February 3 before class (6 points). (19) Calculate explicitly δ x i L ( i = 1 , . . . , 3) for the Lagrangian of the 3D Kepler problem L = m 1 2 ( ˙ ~x 1 ) 2 + m 2 2 ( ˙ ~x 2 ) 2-g m 1 m 2 | ~x 1-~x 2 | Are there associated conservation laws? Due February 3 in class (4 points)....
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