✎ Symplectic Structure [The Physics Travel Guide]

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====== Symplectic Structure ======

<tabbox Why is it interesting?> 
<blockquote>
As each skylark must display its comb, so every branch of mathematics must finally display symplectization. In mathematics there exist operations on different levels: function acting on numbers, operators acting on functions, functors acting on operators, and so on. Symplectization belongs to the small set of highest level operations, acting not on details (functions, operators, functions=, but on all the mathematics at once.

<cite>Catastrophe Theory, by V. Arnold</cite>
</blockquote>

<blockquote>
The word symplectic was coined by Hermann Weyl in his famous treatise The Classical groups [...]

Weyl devoted very little space to the symplectic group, it was then a rather baffling oddity which presumably existed for some purpose, though it was not clear what. **Now we know: the purpose is dynamics.**

In ordinary euclidean geometry the central concept is distance. To capture the notion of distance algebraically we use the inner (or scalar) product $ x.y$ of two vectors $x$ and $y$.  [...] All the basic concepts of euclidean geometry can be obtained from the inner product. [...] The inner product is a bilinear form - the terms look like $x_i y_j$. Replacing it with other bilinear forms creates new kinds of geometry.  Symplectic geometry corresponds to the form $x_1 y_2 -x_2 y_1$, which is the area of the parallelogram formed by the vectors $x$ and $y$. [...] The symplectic form provides the plane with a new kind of geometry, in which very vector has length zero and is at right angles to itself. [...] Can such bizarre geometries be of practical relevance? Indeed they can: they are the geometries of classical mechanics.

In Hamilton's formalism, mechanics systems are described by the position coordinates $q_1,\ldots,q_n$, momentum coordinates $p_1,\ldots,p_n$m  and a function $H$ of these coordinates (nowadays called the hamiltonian) which can be thought of as the total energy. Newtons equations of motion take the elegant form $dq/dt=\partial H/\partial p,$ $dp/dt= -\partial H/\partial q$. When solving Hamilton's equations it is often useful to change coordinates. but if the position coordinates are transformed in some way, then the corresponding momenta must be transformed consistently. Pursuing this idea, it turns out that such transformations have to be the symplectic analogies of rigid euclidean motions.** The natural coordinate changes in dynamics are symplectic.** This is a consequence of the asymmetry in Hamitlon's equations, whereby $dq/dt$ is plus  $\partial H/\partial p$, but $dp/dt$ is minus $\partial H/\partial q$, that minus sign again.

<cite>https://www.nature.com/nature/journal/v329/n6134/pdf/329017a0.pdf</cite>
</blockquote>

<blockquote>
I’ve tried to show you that the symplectic structure on the phase spaces of classical mechanics, and the lesser-known but utterly analogous one on the phase spaces of thermodynamics, is a natural outgrowth of utterly trivial reflections on the process of minimizing or maximizing a function S on a manifold Q.

The first derivative test tells us to look for points with

$$d S = 0$$

while the commutativity of partial derivatives says that

$$d^2 S = 0$$

everywhere—and this gives Hamilton’s equations and the [[equations:maxwell_relations|Maxwell relations]].

<cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</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 "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

* [[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

<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
 
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<tabbox Examples>

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<tabbox History>

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