Timeline for Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?
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| Feb 11, 2010 at 4:18 | comment | added | David Bar Moshe | Sorry for the late comment, a remark on your previous comment (about the boundary conditions). If we consider a full Dirichlet BVP by selecting two arbitrary values theta_i and theta_f on the Kaluza Klein S1, for a given time T we will have a unique geodesic. but the constant of motion representing the charge will not in general have the required value. Using the time reparameterization invariance of the geodesic equation, scale the time by a constant factor inducing a similar scaling on the charge value because it is proportional to time derivatives. Thus we can get the required charge. | |
| Feb 10, 2010 at 16:00 | vote | accept | Theo Johnson-Freyd | ||
| Feb 10, 2010 at 16:00 | history | bounty ended | Theo Johnson-Freyd | ||
| Feb 10, 2010 at 16:00 | comment | added | Theo Johnson-Freyd | No, this is all correct. The trick is that we only need the convex neighborhood in the (n+1)-dimensional space to include (images of, under the flow map) hyperplanes with non-zero charge, which should project diffeomorphically onto the n-dimensional base. This can always be done in small enough neighborhoods. | |
| Feb 10, 2010 at 5:13 | comment | added | Theo Johnson-Freyd | So the Jacobi metric exactly does the trick when there is no magnetic field. But I'm still dubious when there is a magnetic field. Originally I was looking for a nbhd in which I can uniquely solve the BVP with pure Dirichlet boundaries. But from the (n+1)-dimensional perspective, now I want a Dirichlet boundary at one end, and some mixed thing (Dirichlet in the original n coordinates, Neumann in the new one, because I want to fix the charge) at the other. I can solve this if I can solve the Dirichlet problem in an infinite cylinder, but I don't know how to do that. | |
| Feb 8, 2010 at 15:36 | comment | added | David Bar Moshe | I don't think that one should worry. Consider for example an isotropic harmonic oscillator on the x-y plabe together with a constant magnetic field in the z-direction. If we look on this problem in the polar plane coordinates, we see that the angular momentum on the plane and also the angular momentum in the Kaluza-Klein S1 are conserved quantities (the hamiltonian depends only on the momenta) and there is no essential difference between these two angular coordinates. | |
| Feb 7, 2010 at 19:18 | comment | added | Theo Johnson-Freyd | Thanks! I'll certainly look in the Marsden book. So you're saying that there's some metric on $\mathbb R^n \times S^1$ so that geodesic flow there is equivalent to the original mechanics on $\mathbb R^n$? Then, a question: should I worry that the solution for geodesics in general wants only a small patch in $S^1$? | |
| Feb 7, 2010 at 14:35 | history | answered | David Bar Moshe | CC BY-SA 2.5 |