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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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This note was uploaded on 12/29/2011 for the course AST 350 taught by Professor Dion during the Fall '09 term at SUNY Stony Brook.
 Fall '09
 dion

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