A particle of mass m moves in an attractive central potential V(r) = - Boh, where k and B are constants. Assume that the

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A Particle Of Mass M Moves In An Attractive Central Potential V R Boh Where K And B Are Constants Assume That The 1 (85.08 KiB) Viewed 28 times

A particle of mass m moves in an attractive central potential V(r) = - Boh, where k and B are constants. Assume that the angular momentum L of the particle is not zero. (a) Write down the Lagrangian. Show that the angular momentum L of the particle is conserved. (b) Determine the total energy of the system in terms of m, r, r, k, B, and L. What is the kinetic energy term, which is only a function of the radial velocity of the particle? What is the effective potential energy term Vefr(r), which is only a function of radial posi- tion of the particle? (c) Sketch the effective potential Vef() above as a function of r for the following three cases • B<0 • 2 > > 0 • B > 2 For what values of B does a stable circular orbit exist? For what values of B are all orbits bounded? (d) For those values of B which support a stable circular orbit, calculate the radius, ro, of the stable circular orbit in terms of m, k, B, and L. (e) Let r = ro + dr. Derive the equation of motion for radial deviations, Sr(t), assum- ing dr is small. Under what conditions will the perturbed orbit be closed?