# Solutions_Part23 - Goldstein Problem 3-6 Problem GPS 3.13...

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Problem GPS 3.13 If the center of force is located at the origin of an x-y coordinate system, then the orbit is a circle that passes through the origin as, for example, drawn below. This circle has radius a , and its origin is at ( a , 0 x
Problem Goldstein 3-6 (a) Start with the di ff erential equation of the orbit in terms of the variable u = 1 /r , l 2 m u 2 μ d 2 u d θ 2 + u = f ( 1 u ) , (1) where l is the angular momentum, m is the mass of the particle, and u = (2 a cos θ ) 1 . (2) The required derivatives are du d θ = tan θ 2 a cos θ , (3) and d 2 u d θ 2 = 1 2 a · 1 cos θ + 2 tan 2 θ cos θ ¸ . (4) After we substitute Eqs.(2) and (4) into (1), we obtain f ( 1 u ) = l 2 m u 2 1 a cos θ