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Unformatted text preview: Physics 601 Homework 8
Due Friday November 5 € € 1. Consider a two
body system with reduced mass µ and a potential of the form V = −ar− k for a, k > 0 (a,k real). a. Show that circular orbits exist for any k ≠ 2 and find the relationship between the radius r0 and L. b. Linearize the equation of motion for r around r0 and i. Show that stable orbits only exist for k < 2 €
ii. Find the oscillation frequency for fluctuations in r for k<2. €
iii. Find the values of k for which the orbits close. €
€ ass µ and a potential of the form 2. Consider a two
body system with reduced m
V = ar k for a, k > 0 (a,k real). a. Show that circular orbits exist for any k and find the relationship between the radius r0 and L. b. Linearize the equation of motion for r around r0 and i. Find the oscillation frequency for fluctuations in r. ii. Find the values of k for which the orbits close. €
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p2 k
− 2 with k>0 (that is an attractive Coulomb or 3. For the by the Hamiltonian H =
2µ r
gravitational system) show that ˆ
ˆ
a. The Runge
Lenz vector defined by A = p × L − αµr (where r = x / r ) satisfies [ A, H ]PB = 0 . Do this by explicit evaluation of the Poisson bracket. This €
means A is conserved b. The Runge
Lenz vector is in the plane of the orbit€
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4. Show by explicit evaluation that €
a. [ Ax , Ly ]PB = Az b. [ Ax , Ay ] = −2µE Lz where A is the Runge
Lenz vector. If one includes cyclic permutation this yields the Lie algebra discussed in class. € €
5. Consider an elliptical orbit in the Kepler problem ( V ( r) = − k / r ) with energy, E, and €
˙
orbital angular momentum l. Find r−1 , r−2 and r 2 where the bar means time average. You may find the viral theorem helpful but you will need more input then that. €
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6. Suppose one has a “hard sphere” of radius R off of which a particle scatters via specular reflection
that is angle of incidence = angle of reflection on the plane tangent to the sphere. Find the differential cross
section. 7. Consider the potential V ( r) = α / r 2 . Find the differential cross
section. € ...
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 Fall '11
 Hassam
 mechanics, Mass, Orbits, Work

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