410ahw5soln

# 410ahw5soln - HW#5Solutions Phys410Fall 2011 Prof Ted...

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HW#5—Solutions —Phys410—Fall 2011 Prof. Ted Jacobson Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/410a/ 8.2 ( two bodies in an external ﬁeld )) (a) The only new terms that were not in the analysis of (8.13) are the contributions to the potential energy due to the external ﬁeld, U g = - m 1 g · r 1 - m 2 g · r 2 = - M g · R . This containts only R and not r , hence the Lagrangian still separates into two terms. (b) The R equation describes the center of mass motion in the external ﬁeld, and the r equation describes the relative motion which is decoupled from the external ﬁeld. This decoupling is in a sense accidental: the external ﬁeld certainly breaks the symmetry in spatial directions, but since it is proportional to both the masses and the positions, the terms just happen to combine to involve only the center of mass position. (c) If instead of a uniform external ﬁeld we take the ﬁeld of a point mass m 0 , and use that mass as the origin of coordinates, then the extra contributions to the potential energy are U g ( r 1 ,r 2 ) = - Gm 0 m 1 | r 1 | - Gm 0 m 2 | r 2 | . The Lagrangian for the earth-moon system in the ﬁeld of the sun is thus L = 1 2 M ˙ R 2 + 1 2 μ ˙ r 2 + Gm e m m r + Gm s ± m e r e + m m r m ² , where M = m e + m m is the sum of the earth and moon masses, μ = m e m m /M , r e = | R - ( m m /M ) r | , and r m = | R + ( m e /M ) r | . (d)

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410ahw5soln - HW#5Solutions Phys410Fall 2011 Prof Ted...

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