Symplectic geometry & classical mechanics, Lecture 8

For winter semester 2017-18 I am giving a course on symplectic geometry and classical mechanics. This course is intended for anyone with a familiarity with classical mechanics and basic differential geometry.

Here in the 8th lecture, I introduce symplectic vector spaces and manifolds.

[wdlang]

the only course on symplectic geometry for physicists.

[kouhawk]

Indeed very helpful in teaching me symplectic methods, which i came across when learning QFT

[tf]

Sorry, I have a question. What's the motivation to require the symplectic form to be closed? Because from naive thinking, we want some object whose every point's tangent space is like a symplectic vector space, it seems the nondegenerate 2-form is enough. Thanks.

[xwyl]

I feel this course requires a good understanding of differential geometry, for most of the content comes without any explanation.

[reinerwilhelms-tricarico]

I was scratching my head about where to get such a form Q(u, v) = -Q(v, u)
But it's easy: any anti-symmetric matrix will do the job, even though the matrix
won't necessarily be symplectic (why not I'm not sure).

So I was tinkering with trying to get a decomposition of a arbitrary skew symmetric matrix
into three bases uj, ej, fj. So started from a random anti-symmetric matrix, using singular value decomposition (SVD)
Here is my "proof" by programming :-) Octave code (poor man's Matlab):

function [M, uu, ee, ff] = threebasesymplectic (n, m)
% Creates a random symplectic matrix of rank 2*m and dimension d = 2*n+m
% It makes an antisymmetric random matrix M.
% then computes the singular value decomposition such that u*s*v'=M.
% The diagonal s then contains 2*m singular values, which are m different
% values that each appear twice. Each of the m singular values has a
% two-dimensional eigenspace - I suppose simply by virtue of the
% anti-symmetry of M.
% The program then splits the svd system into 3 parts:
% The kernel uu: all the vectors for which the diagonal s of
% the svd is zero, that is, for each vector in uu we get: uu'*M=0;
% Octave's svd already puts those at the end, and the sing. values
% are ordered with the largest first.
% The returned matrices ee and ff contain alternating eigenvectors
% for each of the m 2-dimensional eigenspaces. One of each eigenvector of a pair
% goes to ee, the other to ff.
% In testing you should get: ee'*ee = ff'*ff = Identity (m, m). Further:
% ee*M*ff' or ff'*M*ee gives a diagonal matrix of size m containing the m singular
% values (or their negatives). These are double in the diagonal s from the svd.
% Further: ee'*M*ee and ff'*M*ff are (m, m) zero matrices.
%
d = 2*n+m; % total dimension of the space V
A = randn(d, m); % we want rank 2 m
B = randn(d, m); %
M = A*B'-B*A'; % This is automatically anti-symmetric.
[u, s, v]=svd(M); % singular value decomposition s.t. u*s*v'=M
uu = u(:, 2*m+1:d); % for these the singular values are zero.
ee = u(:, [1:m]*2-1); % alternating 1, 3, 5, ...
ff = u(:, [1:m]*2); % alternating 2, 4, 6, ...
endfunction

(enjoy!)