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HW6_prob3_GPS13_G6

# HW6_prob3_GPS13_G6 - 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 ). The equation that describes it is ( x ± a ) 2 + y 2 = a 2 . In terms of polar coordinates measured from the center of force, we have x = r cos θ and y = r sin θ . After substituting for x and y in the equation of the circle, we obtain the representation of the orbit in polar coordinates as r = 2 a cos θ . Note that a full orbit is traversed as θ varies from ± π /2 to π /2 instead of the usual 0 to 2 π .

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Problem Goldstein 3-6 (a) Start with the di f 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
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HW6_prob3_GPS13_G6 - Goldstein Problem 3-6 Problem GPS 3.13...

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