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HW6_prob4_V=klnr - Two-body central force problem for a...

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Two-body central force problem for a logarithmic potential (a) This is a problem of motion in a central force fi eld determined by the potential V ( r ) = k ln( r/a ) , (1) where k and a are constants. The Lagrangian is L = m 2 μ · r 2 + ( r · θ ) 2 k ln( r/a ) . (2) Since θ is a cyclic variable, ∂L/∂ · θ is a constant of the motion, the angular momentum l , and we can write immediately l = mr 2 · θ . (3) To fi nd the r EOM we fi rst calculate ∂L ∂r = mr · θ 2 k r , (4) which allows us to write the EOM as m ·· r mr · θ 2 + k r = 0 . (5) Using Eq.(3) we can eliminate · θ from Eq.(5) to obtain the equation of motion for the equivalent one-dimensional radial problem m ·· r = f 0 ( r ) . (6) where the e ff ective force is f 0 ( r ) = k r + l 2 mr 3 . (7) Note that the angular momentum l is a parameter whose value may be varied by choosing di ff erent initial values for r and · θ . To use the e ff ective potential V 0 to discuss the qualitative nature of the motion, we need to make qualitative plots of V 0 versus r for di ff erent values of l . Since f 0 ( r ) = dV 0 /dr , we immediately realize that V 0 ( r ) = V ( r ) + l 2 2 mr 2 . (8) Since we do not want our analysis to depend on speci fi c values for a or k , we fi rst introduce a scaled radial coordinate x = r/a . This allows us to write
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