Gromov's nonsqueezing theorem
/ GROH-mof /
Here is a question that sounds like it should be easy. Can you take a round ball of phase space and, using only volume-preserving motions, squeeze it through a narrow opening into a long thin tube of the same volume? For a fluid the answer is obviously yes — stretch it long and thin. But if you must preserve the symplectic structure (as all of mechanics does), Gromov's nonsqueezing theorem says no: there is an invisible obstruction that volume alone cannot see. A symplectic ball simply cannot be squeezed into a cylinder of smaller radius, no matter how long the cylinder.
Precisely, consider in (R^{2n}, omega_0) the ball B(r) of radius r and the cylinder Z(R) = {x_1^2 + y_1^2 < R^2} (a disk of radius R in the first symplectic coordinate plane, times all of R^{2n-2}). Gromov's theorem (1985) states: there exists a symplectic embedding of B(r) into Z(R) if and only if r <= R. The radius of the cylinder — capturing only a single 2-dimensional symplectic cross-section — is an absolute barrier, even though Z(R) has infinite volume and could easily swallow B(r) by a volume-preserving map when r > R. The proof introduced pseudoholomorphic curves into symplectic topology: one finds a J-holomorphic disk through any point of the embedded ball whose symplectic area is bounded below by pi r^2 and above by pi R^2, forcing r <= R. This single argument launched modern symplectic topology.
Why it matters: nonsqueezing is the cleanest proof that symplectomorphisms are strictly more rigid than volume-preserving maps in dimension 2n > 2 — the gap between Symp and the volume-preserving group is real and measurable. It is the origin of symplectic capacities (the Gromov width c(M) = pi r^2 for the largest ball that embeds is the prototypical capacity, monotone and conformal under symplectic maps), and it has a striking physical reading sometimes called the 'symplectic camel': a blob of phase space cannot pass through a hole smaller than its symplectic width, a quantum-uncertainty-like constraint at the purely classical level. A caveat: the statement is specifically about symplectic embeddings into the cylinder Z(R) whose cross-section is a SYMPLECTIC 2-plane; squeezing is possible if the thin directions are paired across complementary (e.g. a q-direction with its own p), so the choice of which 2-plane is the cylinder's cross-section is essential and not interchangeable.
In (R^4, omega_0) with coordinates (x_1, y_1, x_2, y_2), you can never symplectically map the ball B(2) (radius 2) into the cylinder Z(1) = {x_1^2 + y_1^2 < 1}, even though Z(1) has infinite volume and B(2) has finite volume. But you CAN map B(2) into the 'wrong' cylinder {x_1^2 + x_2^2 < 1} of small radius, because that cross-section mixes an x with another x — it is not a symplectic 2-plane (omega_0 restricted to it vanishes), so nonsqueezing does not apply there.
B(r) embeds symplectically in Z(R) iff r <= R — the cross-section must be a symplectic 2-plane.
The obstruction is symplectic, not volumetric: Z(R) has infinite volume, so the barrier r <= R cannot come from volume. And it hinges on the cylinder's cross-section being a symplectic 2-plane — squeeze through a mixed (isotropic) cross-section and the theorem says nothing.