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 __ rr = (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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