Timeline for Difference between manifold and R^n for dynamical systems
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 4, 2017 at 15:59 | comment | added | Thomas Moore | Hi, yes. I guess i'm just confused about the Wikipedia statement: "If the manifold M is locally diffeomorphic to Rn, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional." | |
| Mar 4, 2017 at 15:58 | comment | added | Nadiels | Maybe some confusion about manifolds - $M$ can be finite dimensional! To say that $M$ is an $n$-dimensional manifold is to say that "locally it looks like $\mathbb{R}^n$. Maybe you haven't been exposed to much differential geometry yet? In which case the reference that treats dynamic systems on $\mathbb{R}^n$ is probably the one to work through first. | |
| Mar 4, 2017 at 15:46 | comment | added | Thomas Moore | Hi. Thanks that is very helpful. Is dimensionality a question though? If my dynamical system is on R^n, then, I suppose it is finite-dimensional, so I have a finite number of equations to work with. If my dynamical system is on $M$, the it is infinite-dimensional, and I assume I'm dealing with PDEs? Is that correct? | |
| Mar 4, 2017 at 15:44 | vote | accept | Thomas Moore | ||
| Mar 4, 2017 at 15:33 | comment | added | Nadiels | Absolutely! Imagine any dynamic system you like on the surface of our planet, these would be better considered as dynamic systems on the sphere $S^2$ than as dynamic systems in the ambient space $\mathbb{R}^3$ | |
| Mar 4, 2017 at 15:29 | comment | added | Thomas Moore | Okay, that is helpful, thanks! Is there a situation where $M$ is not R^n? | |
| Mar 4, 2017 at 15:28 | answer | added | Lee Mosher | timeline score: 6 | |
| Mar 4, 2017 at 15:22 | comment | added | Nadiels | They aren't equivalent, rather the first is a particular instance of the more general situation. Both want to say that the dynamic system is governed by a (smoothly varying) vector field that describes the infinitesimal motion of a point $x$ under the flow of $f$, only in the more abstract setting it requires more work to properly define what is meant by the vector field at a point, in both cases the proper place for the vector field to live at each point is $T_x M$, but when $M = \mathbb{R}^n$ the tangent plane at $x$ is isomorphic to $\mathbb{R}^n$ itself. | |
| Mar 4, 2017 at 15:02 | history | asked | Thomas Moore | CC BY-SA 3.0 |