Timeline for Arnold's book on classical mechanics [duplicate]
Current License: CC BY-SA 4.0
13 events
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| Feb 25, 2021 at 23:53 | comment | added | Overflowian | People deal with them sometimes for doing useful constructions but mostly because it's a beautiful formalism. Also another point in adopting the symplectic manifold pov is that is perfect to talk about integrable system -which is now a geometric global property-. Unfortunately quantum mechanics kind of lacks this beautiful geometric background. It is inherently Euclidean. And trying to bridge between classic mechanics and QM brings us to the problem of geometric quantisation (but is not needed for QM to work). | |
| Feb 25, 2021 at 23:44 | comment | added | Overflowian | Yes it is useful to classical mechanics. Arnold's point is that a classical mechanics' problem can be translated in a symplectic geometry problem. If you limit yourself to $R^{2n}$ you can study local properties of the system (eg. what happens near an equilibrium point but you may loose the global properties. For example if you study a spherical pendulum your configuration space wouldn't be $R^{2n}$ so you need to consider more general manifolds. Since any abstract manifold can be realized as a submanifold of $R^N$ you don't need abstract manifolds. continues--- | |
| Feb 25, 2021 at 22:35 | history | closed |
Ben McKay Tyrone Carlo Beenakker Friedrich Knop Francois Ziegler |
Duplicate of Applications of symplectic geometry to classical mechanics | |
| Feb 25, 2021 at 19:32 | comment | added | Jonny Evans | Just to add to alvarezpaiva's comment: 1. when you look at a constrained system you end up working on a symplectic quotient and this is often nonlinear; 2. symplectic quotients also arise in gauge theory (e.g. moduli space of flat connections). | |
| Feb 25, 2021 at 16:46 | answer | added | Alexandre Eremenko | timeline score: 0 | |
| Feb 25, 2021 at 16:05 | comment | added | alvarezpaiva | Even if you start doing classical Hamiltonian mechanics in $R^{2n}$ you may easily end up in some more exotic symplectic manifold if you start to quotient out the symmetries. By the way, the first symplectic manifold to be considered was not $R^{2n}$, but the space of oriented straight lines in $R^3$ (Hamilton's paper on systems of rays). | |
| Feb 25, 2021 at 15:35 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo
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| Feb 25, 2021 at 15:32 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
edited title
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| Feb 25, 2021 at 14:18 | review | Close votes | |||
| Feb 25, 2021 at 22:43 | |||||
| Feb 25, 2021 at 14:16 | comment | added | asv | I am aware of that question. It is useful, but my question is more specialized. | |
| Feb 25, 2021 at 13:42 | history | edited | asv | CC BY-SA 4.0 |
added 46 characters in body
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| Feb 25, 2021 at 13:36 | review | First posts | |||
| Feb 25, 2021 at 14:55 | |||||
| Feb 25, 2021 at 13:32 | history | asked | user174848 | CC BY-SA 4.0 |