Homework 8 - Physics 601 Homework 8   Due Friday...

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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. € € 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€ € € 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. € €€ € 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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