The Kepler Problem Part 1-Classical and Relativistic Mechanics-Lecture Handout, Exercises of Classical and Relativistic Mechanics

Download The Kepler Problem Part 1-Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! Thus the energy looks just like the energy of a particle of mass m in a potential Veff on the half-line {0 < r <∞}. We have reduced the problem to a 1-dimensional problem! Veff is called the effective potential. Note that the second term creates the effect of a repulsive force equal to j2/mr3, called the centrifugal force. 3. Show that ṙ = √ 2 m (E − Veff(r)). (4) We could solve this differential equation to find r as a function of t, but it’s nicer to find r as a function of θ, since this allows us to see the shape of the particles’ orbits. In fact it turns out to be easier to first find θ as a function of r and then solve for r in terms of θ — so that’s what we’ll do. 4. Using equations (3) and (4) show that dθ dr = j/mr2√ 2 m (E − Veff (r)) . Conclude that θ = θ0 + ∫ (j/mr2) dr√ 2 m (E − Veff(r)) (5) Now let’s specialize to the case of gravity, where f(r) = −k/r2 and thus V (r) = −k/r for some constant k. 5. Sketch a graph of the effective potential Veff(r) in this case, and say what a particle moving in this potential would do, depending on its energy E. 6. Show using equation (5) that θ = θ0 + arccos j mr − kj√ 2E m + k2 j2 . This is the only part of this homework where you really need to sweat. However, some ways to do it are easier than others, so think a bit before you plunge into an enormous masochistic calculation — if you do it intelligently, you will only need a medium-sized masochistic calculation! For example, you may want to derive a general formula for ∫ dx√ ax2 + bx+ c and then use it to do the integral in equation (5). 7. Reduce the clutter a bit more by defining p = j2/km, e = √ 1 + 2Ej2 mk2 . Show that in terms of these variables we have θ = θ0 + arccos ( p/r − 1 e ) 2 docsity.com