Lagrangian submanifolds, symplectic area, and Gromov's non-squeezing theorem (1985)
Phase Space & Symplectic Form
Symplectic form: ω = dp ∧ dq (area form on phase space). Liouville's theorem: Hamiltonian flow preserves ω — symplectic maps are area-preserving.
The Lagrangian condition: L ⊂ (M²ⁿ, ω) with dim L = n and ω|_L = 0.
Gromov Non-Squeezing
Gromov Non-Squeezing (1985): A symplectic ball B²ⁿ(r) can be embedded into the cylinder Z²ⁿ(R) = B²(R) × ℝ²ⁿ⁻² symplectically iff r ≤ R.
This is the first rigidity result: symplectic maps are much more rigid than volume-preserving maps (which can always squeeze).