ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 3 (January 21, 2011)

(11a) Assume a bilinear kinetic Energy

T = ∑_{j,k} a_{jk} q̇_{j}q̇_{k} and prove ∑_{i} q̇_{i} \frac{∂T}{∂q̇_{i}} = 2T .

Due in class (2 points). (11b) Generalized Momentum:

Calculate \frac{∂L}{∂ẋ_{i}} , i = 1, 2, 3, for L = \frac{1}{2} m v^{2} − V (x⃗).

Due in class (2 points).

(12) Legendre transformation: Define the Hamiltonian by

H = (∑_{j} q̇_{j} \frac{∂L}{∂q̇_{j}} − L) and the generalized momentum by p_{j} = \frac{∂L}{∂q̇_{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 26 before class (10 points).