ay45c4-page7

# ay45c4-page7 - That is the angular momentum vector is xed...

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Unformatted text preview: That is, the angular momentum vector is xed in direction and magnitude (because no external torques are acting). L M1 r1 r2 r1 CoM r2 M2 The angular momentum provides 3 more integrals of the motion, so we are now up to 9. We can also look at the total energy of the system: E = (kinetic energy) + (potential energy)   1 1 M r  r + GM1M2 : __ = 2 M1r1  r1 + 2 2 _ 2 _ 2 jr1 r2j To see that this is conserved, write dE = M r  r + M r  r + GM1 M2 d jr 1 _ 1 1 2 _ 2 2 dt jr1 r2j2 dt 1 r 2j : Now d jr dt 1 Therefore r2j d = dt (r1  r1 + r2  r2 2r 1  r2)1=2 _ _ __ rr _ = r1  r1 + r2 jr2 r1j r2 r1  r2 = (r1 jr 2)(_r1 j r2) : r1 r2 1 2  dE = r M r + GM1 M2 (r _ 1 1 1 jr r j3 1 dt 1 2 =0   _  GM r r2) + r2 M2r2 + jr 1 M2j3 (r2 1 2 using the equations of motion. Therefore the total energy of the system, E = constant 66 r 1)  ...
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