Unformatted text preview: (14) Derive (due January 25 in class, 8 points) d dt L-X j ˙ q j ∂L ∂ ˙ q j = 0 from δ t L = ∂L ∂t δt = 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 Hamilton’s equations of motion. Hint: Calculate dH . Due January 29 in class (6 points)....
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- Spring '11
- Work, pj, Lagrangian mechanics, SMALL OSCILLATIONS, generalized momentum, bilinear kinetic Energy