11msolution - p we obtain the Hamiltonian H = T V = p 2 2 m...

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ADVANCED DYNAMICS — PHY–4241/5227 Midterm Exam Solutions PROBLEM 1 From all possible paths that a physical system can take in going from point x 1 at time t 1 to point x 2 at time t 2 it will select the one that minimizes the action S = Z t 2 t 1 dtL = Z t 2 t 1 dt ( T - V ) under local variations of the path. The Euler-Lagrange equations follow. In our case: d dt ∂L ˙ x = ∂L ∂x m ¨ x = - V 0 ( x ) = F ( x ) in agreement with Newton’s second law. PROBLEM 2 (1) The momentum is p = ∂L ˙ x = m ˙ x (2) Eliminating ˙ x in favor of
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Unformatted text preview: p we obtain the Hamiltonian H = T + V = p 2 2 m + 1 2 kx 2 (3) Hamilton’s equations are ∂H ∂p = p m = ˙ x and ∂H ∂x = kx =-˙ p (4) Newton’s force law follows kx =-˙ p =-m ¨ x or m ¨ x =-kx PROBLEM 3 The given Lagrangian is independent of t and φ . Thus, ∂L ∂t = 0 ⇒ E = T + V conserved . and 0 = ∂L ∂φ = d dt ∂L ∂ ˙ φ = mr 2 sin 2 ( θ ) ˙ φ = p φ conserved ....
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This note was uploaded on 11/10/2011 for the course PHY 4241 taught by Professor Berg during the Spring '11 term at University of Florida.

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