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advanced_tools:symplectic_structure [2018/02/14 12:37]
86.151.159.144 [Layman]
advanced_tools:symplectic_structure [2018/10/11 14:59] (current)
jakobadmin [Concrete]
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 ====== Symplectic Structure ====== ====== Symplectic Structure ======
 +
 +
 +<tabbox Intuitive> ​
 +<​blockquote>​Our everyday world is ruled by Euclidean geometry (and by its extension,
 +Riemannian geometry); we can measure distances in it, and velocities. Far
 +away from our daily experience, and much more subtle, is the mechanical
 +phase space world, in which all the phenomena related to simultaneous consideration
 +of position and variation of position; a deep understanding of
 +this world requires the recourse to a somewhat counter-intuitive geometry, the symplectic geometry of Hamiltonian mechanics. Symplectic geometry
 +is highly counter-intuitive;​ the notion of length does not make sense there,
 +while the notion of area does. This "​areal"​ nature of symplectic geometry,
 +which was not realized until very recently, has led to unexpected mathematical
 +developments,​ starting in the mid 1980's with Gromovís discovery of a
 +"​non-squeezing"​ phenomenon which is reminiscent of the quantum uncertainty
 +principle—but in a totally classical setting! <​cite>​[[https://​www.univie.ac.at/​nuhag-php/​bibtex/​open_files/​7041_PhysRepSubmissionGossonLuef.pdf|Symplectic Capacities and the Geometry of Uncertainty]] by Maurice de Gosson et. al. </​cite></​blockquote>​
 +
 +
 +  * [[https://​www.quantamagazine.org/​the-fight-to-fix-symplectic-geometry-20170209?​utm_content=buffer1e5eb&​utm_medium=social&​utm_source=twitter.com&​utm_campaign=buffer|A Fight to Fix Geometry’s Foundations]] by Kevin Hartnett
 +
 +<tabbox Concrete> ​
 +<​blockquote>​A simple way of putting it is that a two-form a way of measuring area in multivariable calculus. I believe the significance for physics boils down to the following: it turns out that a two-form is precisely what is required to translate an energy functional on phase space (a Hamiltonian) into a flow (a vector field). [See Wikipedia for how the translation goes, or read Arnold'​s book Mathematical Methods of Classical Mechanics, or a similar reference.] The flow describes time evolution of the system; the equations which define it are Hamilton'​s equations. One property these flows have is that they preserve the symplectic form; this is just a formal consequence of the recipe for going from Hamiltonian to flow using the form. So, having contemplated momentum, here we find ourselves able to describe how systems evolve using the phase space T*M, where not only is there an extremely natural extra structure (the canonical symplectic form), but also that structure happens to b preserved by the physical evolution of the system. That's pretty nice! Even better, this is a good way of expressing conservation laws. When physical evolution preserves something, that's a conservation law. So in some sense, "​conservation of symplectic form" is the second most basic conservation law. (The most basic is conservation of energy, which is essentially the definition of the Hamiltonian flow.) You can use conservation of symplectic form to prove the existence of other conserved quantities when your system is invariant under symmetries (this is Noether'​s theorem, which can also be proved in other ways, I think, but they probably boil down to the same argument ultimately). <​cite>​http://​qr.ae/​TUTIn9</​cite></​blockquote>​
 +
 +----
 +
 +  * [[http://​math.mit.edu/​~cohn/​Thoughts/​symplectic.html|Why symplectic geometry is the natural setting for classical mechanics]] by Henry Cohn
 +  * Chapter "​[[https://​link.springer.com/​chapter/​10.1007%2F978-1-4612-0189-2_3#​page-1|Phase Spaces of Mechanical Systems are Symplectic Manifolds"​ in the book  Symmetry in Mechanics]] by Stephanie Frank Singer
 +  * See also Chapter 1 in Principles Of Newtonian And Quantum Mechanics by Gosson
 +
 +
 + 
 +<tabbox Abstract> ​
 +
 +<​blockquote>"​The symplectic geometry arises from the understanding of the fact that the transformations of the phase flows of the dynamical systems of classical mechanics and of variational calculus ~and hence also of optimal control theory! belong to a narrower class of diffeomorphisms of the phase space, than the incompressible ones. Namely, they preserve the so-called symplectic structure of the phase space—a closed nondegenerate differential two-form. This form can be integrated along two-dimensional surfaces in the phase space. The integral, which is called the Poincare´ integral invariant, is preserved by the phase flows of Hamilton dynamical systems. The diffeomorphisms,​ preserving the symplectic structure—they are called symplectomorphisms—form a group and have peculiar geometrical and topological properties. For instance, they preserve the natural volume element of the phase space ~the exterior power of the symplectic structure 2-form! and hence cannot have attractors"​
 +
 +<​cite>​[[http://​aip.scitation.org/​doi/​pdf/​10.1063/​1.533315|Symplectic geometry and topology]] by V. I. Arnold</​cite></​blockquote>​
 +--> What's the relation to the symplectic groups?#
 +
 +  * See http://​math.ucr.edu/​home/​baez/​symplectic.html
 + 
 +<--
  
 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
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 $$d^2 S = 0$$ $$d^2 S = 0$$
  
-everywhere—and this gives Hamilton’s equations and the [[equations:​maxwell_relations|Maxwell relations]]. ​+everywhere—and this gives Hamilton’s equations and the [[formulas:​maxwell_relations|Maxwell relations]]. ​
  
 <​cite>​https://​johncarlosbaez.wordpress.com/​2012/​01/​23/​classical-mechanics-versus-thermodynamics-part-2/</​cite>​ <​cite>​https://​johncarlosbaez.wordpress.com/​2012/​01/​23/​classical-mechanics-versus-thermodynamics-part-2/</​cite>​
 </​blockquote>​ </​blockquote>​
  
-**Further Reading:**+<​blockquote>​Hamilton'​s equations push us toward the viewpoint where $p$ and $q$ have equal status as coordinates on the phase space $X$.  Soon, we'll drop the requirement that $X\subseteq T^\ast Q$ where $Q$ is a configuration space. ​ $X$ will just be a manifold equipped with enough structure to write down Hamilton'​s equations starting from any $H \colon X\rightarrow\mathbb{R}$.
  
-  * [[http://​www.pims.math.ca/​~gotay/​Symplectization(E).pdf|THE SYMPLECTIZATION OF SCIENCE]] by Mark J. Gotay et. al+The coordinate-free description of this structure is the major 20th century contribution to mechanicsa symplectic structure.
  
 +This is important. ​ You might have some particles moving on a manifold like $S^3$, which is not symplectic. ​ So the Hamiltonian mechanics point of view says that the abstract manifold that you are really interested in is something different: it must be a symplectic manifold. ​ That's the phase space $X$. <​cite>​[[http://​math.ucr.edu/​home/​baez/​classical/​texfiles/​2005/​book/​classical.pdf|Lectures on Classical Mechanic]]s by J. Baez</​cite></​blockquote>​
  
 +<​blockquote>​The mathematical structure underlying both classical
 +and quantum dynamical behaviour arises from symplectic geometry. It turns
 +out that, in the quantum case, the symplectic geometry is non-commutative,​
 +while in the classical case, it is commutative.<​cite>​https://​arxiv.org/​pdf/​1602.06071.pdf</​cite></​blockquote>​
  
 +**Further Reading:**
  
-<tabbox Layman>  +  * [[http://www.pims.math.ca/~gotay/Symplectization(E).pdf|THE SYMPLECTIZATION OF SCIENCE]] by Mark J. Gotay et. al. 
-<​blockquote>​Our everyday world is ruled by Euclidean geometry (and by its extension,​ +
-Riemannian geometry); we can measure distances in it, and velocities. Far +
-away from our daily experience, and much more subtle, is the mechanical +
-phase space world, in which all the phenomena related to simultaneous consideration +
-of position and variation of position; a deep understanding of +
-this world requires the recourse to a somewhat counter-intuitive geometry, the symplectic geometry of Hamiltonian mechanics. Symplectic geometry +
-is highly counter-intuitive;​ the notion of length does not make sense there, +
-while the notion of area does. This "​areal"​ nature of symplectic geometry, +
-which was not realized until very recently, has led to unexpected mathematical +
-developments,​ starting in the mid 1980's with Gromovís discovery of a +
-"​non-squeezing"​ phenomenon which is reminiscent of the quantum uncertainty +
-principle—but in a totally classical setting! <​cite>​[[https://www.univie.ac.at/nuhag-php/bibtex/​open_files/​7041_PhysRepSubmissionGossonLuef.pdf|Symplectic Capacities and the Geometry of Uncertainty]] by Maurice de Gosson ​et. al. </​cite></​blockquote>​+
  
- 
-  * [[https://​www.quantamagazine.org/​the-fight-to-fix-symplectic-geometry-20170209?​utm_content=buffer1e5eb&​utm_medium=social&​utm_source=twitter.com&​utm_campaign=buffer|A Fight to Fix Geometry’s Foundations]] by Kevin Hartnett 
- 
-<tabbox Student> ​ 
- 
-  * [[http://​math.mit.edu/​~cohn/​Thoughts/​symplectic.html|Why symplectic geometry is the natural setting for classical mechanics]] by Henry Cohn 
-  * Chapter "​[[https://​link.springer.com/​chapter/​10.1007%2F978-1-4612-0189-2_3#​page-1|Phase Spaces of Mechanical Systems are Symplectic Manifolds"​ in the book  Symmetry in Mechanics]] by Stephanie Frank Singer 
- 
- 
-  
-<tabbox Researcher> ​ 
- 
-<​blockquote>"​The symplectic geometry arises from the understanding of the fact that the transformations of the phase flows of the dynamical systems of classical mechanics and of variational calculus ~and hence also of optimal control theory! belong to a narrower class of diffeomorphisms of the phase space, than the incompressible ones. Namely, they preserve the so-called symplectic structure of the phase space—a closed nondegenerate differential two-form. This form can be integrated along two-dimensional surfaces in the phase space. The integral, which is called the Poincare´ integral invariant, is preserved by the phase flows of Hamilton dynamical systems. The diffeomorphisms,​ preserving the symplectic structure—they are called symplectomorphisms—form a group and have peculiar geometrical and topological properties. For instance, they preserve the natural volume element of the phase space ~the exterior power of the symplectic structure 2-form! and hence cannot have attractors"​ 
- 
-<​cite>​[[http://​aip.scitation.org/​doi/​pdf/​10.1063/​1.533315|Symplectic geometry and topology]] by V. I. Arnold</​cite></​blockquote>​ 
---> What's the relation to the symplectic groups?# 
- 
-  * See http://​math.ucr.edu/​home/​baez/​symplectic.html 
-  
-<-- 
- 
---> Common Question 2# 
- 
-  
-<-- 
-  ​ 
-<tabbox Examples> ​ 
- 
---> Example1# 
- 
-  
-<-- 
- 
---> Example2:# 
- 
-  
-<-- 
-  ​ 
-<tabbox History> ​ 
  
 </​tabbox>​ </​tabbox>​
  
  
advanced_tools/symplectic_structure.1518608248.txt.gz · Last modified: 2018/02/14 11:37 (external edit)