HW13_prob6_GPS27_G17

HW13_prob6_GPS27_G17 - GPS 8-27(2nd ed 8.17(a Using the...

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GPS 8-27 (2 nd ed. 8.17) (a) Using the identity, sin 2 α =2s in α cos α , the given Lagrangian can be rewritten as L = m 2 · q 2 sin 2 ωt +2 · qqω sin ωt cos ωt + ω 2 q 2 ¸ , (1) from which we f nd the generalized momentum p p = ∂L · q = m · q sin 2 ωt + qmω sin ωt cos ωt . (2) The energy function h is h = · q ∂L · q L = m 2 · q 2 sin 2 ωt ω 2 q 2 ¸ , (3) and using Eq.(2) to substitute for · q we f nd several equivalent forms for the Hamiltonian H = 1 2 m h ( p qmω sin ωt cos ωt ) 2 sin 2 ωt m 2 ω 2 q 2 i , (4) or H = 1 2 m h ( p csc ωt ) 2 2 pqmω cot ωt m 2 ω 2 q 2 sin 2 ωt i , (5) or H = 1 2 m h ( p csc ωt qmω cos ωt ) 2 m 2 ω 2 q 2 i . (6) Since H is explicitly a function of time, it is not conserved. (b) Now we introduce a new coordinate Q de f ned as Q = q sin ωt . (7) The generalized velocity is · Q = · q sin ωt + cos ωt , (8) from which it follows that · Q 2 = · q 2 sin 2 ωt +2 · qqω sin ωt cos ωt +
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

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HW13_prob6_GPS27_G17 - GPS 8-27(2nd ed 8.17(a Using the...

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