Module V·Article II·~1 min read
Symplectic Geometry and Its Applications
Affine and Projective Transformations
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Introduction to Symplectic Geometry
Symplectic Form
The symplectic form on $\mathbb{R}^{2n}$: $\omega = \sum_i dp_i \wedge dq_i$ is a nondegenerate skew-symmetric bilinear form.
Symplectic geometry studies spaces with such a form. The key property: $\omega$ is nondegenerate (determinant of the form’s matrix $\neq 0$) and closed ($d\omega = 0$).
Hamiltonian Mechanics
The phase space $(p, q)$ of a mechanical system is a symplectic space. The Hamilton equations: $\dot{q}_i = \partial H / \partial p_i$, $\dot{p}_i = -\partial H / \partial q_i$ preserve the symplectic form (Liouville’s theorem: the volume in phase space is preserved).
The energy $H$ is conserved along a trajectory (if it does not explicitly depend on time).
Poisson Brackets
${f, g} = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)$.
${q_i, p_j} = \delta_{ij}$, ${q_i, q_j} = {p_i, p_j} = 0$ are the canonical Poisson brackets.
Quantization: Poisson brackets are replaced by operator commutators: $[\hat{q}, \hat{p}] = i\hbar$ — the uncertainty principle.
Darboux’s Theorem
On any symplectic manifold, there exist local coordinates $(p, q)$ in which the form takes the standard form. Symplectic geometry is “locally trivial” — in contrast to Riemannian geometry (where curvature is essential).
Symplectic invariants are global. Gromov’s nonsqueezing theorem (1985): a ball in phase space cannot be symplectomorphically embedded into a “narrow cylinder.”
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