Classical Mechanics/Symplectic Spaces

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What makes the phase space variables "momentum" and "position" (p, q) so special, compared to other possible choices such as the set of variables "velocity" and "position"? The reason is the additional property of the phase space (p,q) of having a "symplectic structure".

Let us define a "symplectic unit matrix"

where -1 is an even-dimensional unit matrix. The dimension of the matrices we are talking about is the phase-space dimension, which is always even (each position variable is paired with a momentum variable). This is essentially the matrix generalization of the imaginary unit that we know from complex algebra. Simply put, symplectic spaces arise when unit matrices are replaced by symplectic matrices in strategically chosen places.

This sounds simple, but there is a catch: what is the square root of a matrix? As we already know from complex algebra, there isn't just one single answer for the square root. In mechanics, the root that we mean by I is moreover 'totally antisymmetric'. This still does not completely determine what I is. To get further, we need to think about the physics in phase space. In particular, we will encounter the phenomenon of volume preservation under temporal evolution.