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Symplectic Geometry

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.

Catastrophe Theory, by V. Arnold

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 if 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.

symplectic_geometry.txt · Last modified: 2017/06/28 10:40 by jakobadmin