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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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- Spring '11