Symplectic Structure [The Physics Travel Guide]

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--- advanced_tools:symplectic_structure [2018/02/14 12:29]
86.151.159.144 [Why is it interesting?]
+++ advanced_tools:symplectic_structure [2018/04/15 11:40]
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 ====== 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>
+  * [[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 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?>
@@ Line 37: / Line 73: @@
 </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 mechanics: a 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>
+**Further Reading:**
+  * [[http://www.pims.math.ca/~gotay/Symplectization(E).pdf|THE SYMPLECTIZATION OF SCIENCE]] by Mark J. Gotay et. al.
-<tabbox Layman>
-<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 ìarealî nature of symplectic geometry,
-which was not realized until very recently, has led to unexpected mathematical
-developments, starting in the mid 1980ís with Gromovís discovery of a
-ìnon-squeezingî 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>

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