= Symplectic geometry
{wiki}
A <symplectic form> is a closed, nondegenerate <differential two-form> $\omega$ on a <smooth manifold>. Nondegeneracy forces even <dimension> and converts the <differential> of a <function> into a <Hamiltonian vector field>. For example, with $\omega=\sum_jdq_j\wedge dp_j$ and the convention $\iota_{X_H}\omega=dH$, the <Hamiltonian flow> satisfies
$$
\dot q_j=\frac{\partial H}{\partial p_j},\qquad\dot p_j=-\frac{\partial H}{\partial q_j}.
$$
Thus the geometry of <phase space> encodes the equations of <classical mechanics>. The <cotangent bundle> supplies this structure intrinsically, without a choice of coordinates.<Lectures on the Geometry of Quantization by Sean Bates and Alan Weinstein>[Chapter 3, symplectic structures, cotangent bundles and mechanics]{ref}
The <Symplectic Darboux theorem> makes every such form locally standard; there is no local curvature invariant analogous to <Riemann curvature tensor>. Global geometry still imposes strong constraints. A <symplectomorphism> preserves <symplectic volume>, while the <Gromov non-squeezing theorem> obstructs squeezing a ball through a narrow conjugate coordinate plane even when the target has infinite volume. <Lagrangian submanifolds>, <symplectic reduction> and <J-holomorphic curves> describe different ways this global rigidity interacts with dynamics and <topology>.
Back to article page
