Timeline for Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2016 at 7:52 | comment | added | Ali Taghavi | As you said, the sympletcic structure is coming from the cotangent bundle via isomorphism between the tangent and cotangent bundle via inner product inheriting from $\mathbb{R}^{n+1}$. But before going to the Hamiltonian vector field, please consider your equation $H=1+2H_{\text{geod}}$. The left part is a globally bounded function but the right part is not. How can they be equal? | |
| Dec 17, 2016 at 2:09 | comment | added | Richard Montgomery | sorry, edit system cut me off; 1) you have a point, about bounded/unboundednss. (2) what is your symplectic form, or your Ham vector field on the tangent bundle of the sphere? | |
| Dec 17, 2016 at 1:49 | comment | added | Richard Montgomery | Ali , what symplectic form did you want to use on the tangent bundlewould you write down, in some coordinates, what your symplectic form is, or what your vector field is? The reason I ask -- $TS^n$ does not have an intrinsic symplectic form, but $T^* S^n$ does. I was assuming you were using the standard Riemannian metric to identify the tangent and cotangent bundles in your question. | |
| Dec 13, 2016 at 12:57 | comment | added | Ali Taghavi | Prof. Montgomery Thank you very much for your answer and your attention to my question. I am sorry for my delay. May be I am missing some thing but the Hamiltonian $H$ which I defined is a bounded function but the geodesic Hamiltonian is not. So, how they differ by a constant? | |
| Dec 4, 2016 at 6:56 | history | answered | Richard Montgomery | CC BY-SA 3.0 |