Non-squeezing theorem

The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry.^[1] It was first proven in 1985 by Mikhail Gromov.^[2] The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The theorem is important because formerly very little was known about the geometry behind symplectic maps. One easy consequence of a transformation being symplectic is that it preserves volume.^[3] One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture squeezing the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving.