a1003 - (14) Derive (due January 25 in class, 8 points) d...

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ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 3 (January 20, 2010) (13) A double pendulum consists of two simple pendula, with one pendulum sus- pended from the bob of the other. Assume that the two pendula have equal lengths, have bobs of equal mass and are conFned to move in the same plane. 1. DeFne angles φ and ψ for the pendula with respect to the gravity direction and write down the Lagrange function. 2. Derive the equations of motion for small oscillations around the rest posi- tion φ = ψ = 0. (Small oscillations neglect all terms in the Taylor expan- sion of the Lagrangian, which are higher than quadratic in combinations of φ, ˙ φ, ψ, ˙ ψ .) 3. Calculate the normal modes. Due January 25 before class (10 points).
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Unformatted text preview: (14) Derive (due January 25 in class, 8 points) d dt L-X j q j L q j = 0 from t L = L t t = 0 . Assume a bilinear kinetic Energy T = X j,k a jk q j q k and prove X i q i T q i = 2 T . (15a) Generalized Momentum: Calculate L x i , i = 1 , 2 , 3 , for L = 1 2 m ~v 2-V ( ~x ) . Due January 29 in class (2 points). (15b) Legendre transformation: DeFne the Hamiltonian by H = X j q j L q j-L and the generalized momentum by p j = L q j . Show that the Hamiltonian is a function of q j and p j only: H = H ( q j , p j ). Then derive Hamiltons equations of motion. Hint: Calculate dH . Due January 29 in class (6 points)....
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