solution4ttt - Solution to Problem Set 4 Solution: a)...

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Unformatted text preview: Solution to Problem Set 4 Solution: a) Express the kinetic energy T in terms of the given coordinates: T cm = 1 2 2 m ( r 2 + r 2 2 ) , (1) T = T cm + 1 2 X i m i r 2 i 2 i = m ( r 2 + r 2 2 ) + mL 2 + 2 . (2) b) The Lagrangian of the system: L = T- U, (3) U =- GMm r 2 + L 2 + 2 rL cos +- GMm r 2 + L 2- 2 rL cos , (4) where M is the suns mass. c) Since L/ = 0, p is the constant of motion: p = L = 2 mr 2 + 2 mL 2 + = const. (5) d) Using Euler-Lagrange equation, its straightforward to write down those equations of motion: for r : 2 r = 2 r 2- GM r + L cos ( r 2 + L 2 + 2 rL cos ) 3 / 2 + r- L cos ( r 2 + L 2- 2 rL cos ) 3 / 2 ! , (6) for : 2 r r + ( r 2 + L 2 ) =- L 2 , (7) for : 2 L 2 =- 2 L 2 - GMrL sin 1 ( r 2 + L 2 + 2 rL cos ) 3 / 2- 1 ( r 2 + L 2- 2 rL cos ) 3 / 2 ! . (8) e) Find the Hamiltonian of the system and Hamiltonians equations of motion: p r = L r = 2 m r, (9) p = L = 2 mr 2 + 2 mL 2 + , (10) p = L = 2 mL 2 + , (11) 1 and therefore, H = X i p i q i- L = p 2 r 4 m + ( p - p ) 2 4 mr 2 + p 2 4 mL 2 +- GMm r 2 + L 2 + 2 rL cos +- GMm r 2 + L 2- 2 rL cos . (12) Hamiltonians equations of motion: p i = H q i , q i =- H p i , (13) thus: r = p r 2 m , (14) p r = ( p - p ) 2 2 mr 3- GMm r + L cos ( r 2 + L 2 + 2 rL cos ) 3 / 2 + r- L cos ( r 2 + L 2- 2 rL cos ) 3 / 2 ! ; (15) = p - p 2 mr 2 , (16) p = 0; (17) = p - p 2 mr 2 + p 2 mL 2 , (18) p = GMmrL sin 1 ( r 2 + L 2 + 2 rL cos ) 3 / 2- 1 ( r 2 + L 2- 2 rL cos ) 3 / 2 ! . (19) f) We need the initial conditions r,,, r, , t =0 to solve the dynamical system r = R- L, = = 0 , (20) r t =0 = 0 . (21) According to the previous discussions, p [Eq. (5)] is the total angular momentum of the system, which is conserved. We have p = 2 m ( R- L ) 2 t =0 + 2 mL 2 t =0 + t =0 = m ( V e R + V v ( R- 2 L )) . (22) Additionally, according to our coordinate system: V v- V e = 2 L ( t =0 + t =0 ) , (23) where V v = r GM R- 2 L ,V e = r GM R . (24) Therefore using Eqs. (22) and (23), we find t =0 = V e ( R + L ) + V e ( R- 3 L ) 2 ( R- L ) 2 , (?? TW) (25) 2 t =0 = ( V e + V v ) / (2 r ) (TW) (26) t =0 = V v- V e 2 L- V e ( R + L ) + V e ( R- 3 L ) 2 ( R- L ) 2 . (?? TW) (27) t =0 =- + 1 2 ( V v- V e ) /L (TW) (28) g) see next page (TW) 3 solution to problem set 4 part g The question asks you to discuss whether the motion is confined to small oscillations of or whether the bridge rotates and increases indefinitely. I wanted you to try to formulate a strategy like the one below.increases indefinitely....
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This note was uploaded on 03/02/2010 for the course PHYSICS 316 taught by Professor Pucker during the Spring '10 term at King's College London.

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solution4ttt - Solution to Problem Set 4 Solution: a)...

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