Symplectic applet

by Greg Egan

The applet below applies a symplectic (area-preserving) map to a disk of radius R: taking each point with polar coordinates (r, θ) and mapping it to (√(R2r2), θ).

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You can drag anywhere on the source image to move the location of the disk, or drag on the perimeter of the disk to resize it.

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How can we check that the function:

fR(r, θ) = (√(R2r2), θ)

preserves areas? In polar coordinates, area is measured by:

dA = r dr

So we have:

r2 = √(R2r12)
θ2 = θ1
dA2 = r2 dr22
= √(R2r12) d(√(R2r12)) dθ1
= √(R2r12) (–2r1) / (2 √(R2r12)) dr11
= –r1 dr11 = –dA1

The minus sign tells us that the function fR produces a “mirror-reversed” image, but that the absolute value of the area of any region is unchanged.

One interesting application of this area-preserving property is to show that the volume of the unit-radius ball in 2n dimensions is:

V(B2n) = πn / n!

That is to say, the volume is 1/n! times the volume of the Cartesian product of n unit disks. This is reminiscent of the fact that a hypercube can be dissected into n! simplexes of equal volume. If the hypercube has vertices (±1, ±1, ..., ±1) we can take one simplex to have the n+1 vertices:

v0 = (1, 1, 1, ..., 1)
v1 = (–1, 1, 1, ..., 1)
v2 = (–1, –1, 1, .., 1)
vn = (–1, –1, –1, ..., –1)

and then generate the vertices of all n! simplexes by permuting these coordinates. The simplexes themselves are the regions that satisfy:

–1 ≤ x1x2 ≤ ... ≤ xn ≤ 1

for all n! permutations of the coordinates xi.

If we take a point that lies within the unit ball B2n to be given by polar coordinates (ri, θi) in n planes, we can define the following points in the unit disk:

d1 = f1(r1, θ1), so |d1|2 = 1 – r12
d2 = f|d1|(r2, θ2), so |d2|2 = 1 – r12r22
d3 = f|d2|(r3, θ3), so |d3|2 = 1 – r12r22r23

Clearly the |di| will be in descending order, so we are mapping the unit ball into a region with just 1/n! of the total volume of the product of the unit disks (since every one of the n! possible orders for these magnitudes will occupy an equal volume). Because fR preserves areas, the unit ball must have the same volume as the region into which it is mapped. So we have shown that:

V(B2n) = πn / n!

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Science Notes / Symplectic applet / created Thursday, 15 August 2019
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