ay45c4-page11

# ay45c4-page11 - with an energy equation 1(x2 y2 GM = M E =...

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Unformatted text preview: with an energy equation 1 (x2 + y2) GM = M E = E : __ 2 (x2 + y2)1=2 M1M2  In principle the above two equations are the coupled di erential equations for the two unknowns x(t) and y(t). It is simpler, however, to exploit the planar geometry in polar coordinates r and . r(t + dt) r δθ δr radial direction ^ r δr r (t) azimuthal angle direction ^ θ δθ particle trajectory 0 If r(t) ! r(t + dt) in time dt, then, to rst order, (component of r along original radial direction) = r (component of r along direction of azimuth) = r : Therefore if, at any given time, in cylindrical coordinates (r; ; z), r _ r _ r r __ rr = (r; 0; 0)  rer r(t + t) r(t) =  r ; r  ; 0 = r_er + r_e = t t t _ = 0; 0; r2  = r2 + r2_2 _ 70 ...
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• Fall '09
• dion
• Spherical coordinate system, Polar coordinate system, Cylindrical coordinate system, radial direction, azimuthal angle direction, original radial direction

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