Geometry of closed non-degenerate 2-forms on even-dimensional manifolds. Canonical phase-space setting for Hamiltonian mechanics and the substrate of mirror symmetry; Darboux theorem says all symplectic manifolds are locally identical, so…
symplectic-geometry
Symplectic form
Closed, non-degenerate 2-form ω on a 2n-manifold: dω = 0 and ω^n ≠ 0 pointwise. Non-degeneracy makes T_pM → T_p*M, v ↦ ι_v ω an…
Darboux theorem
Every symplectic manifold (M^{2n}, ω) is locally symplectomorphic to (ℝ^{2n}, Σ dpᵢ∧dqᵢ). Consequence: no local invariants exist — all…
Hamiltonian vector field
Given H ∈ C^∞(M), the unique vector field X_H determined by ι_{X_H}ω = dH. Flow of X_H preserves both ω (Liouville) and H (energy…
Poisson bracket
Bilinear, skew, Leibniz, Jacobi-satisfying bracket on C^∞(M) induced by ω: {f, g} = ω(X_f, X_g) = X_g(f) = –X_f(g). Quantisation replaces…
Lagrangian submanifold
Submanifold L ⊂ (M^{2n}, ω) of dimension n on which ω vanishes. Graphs of symplectomorphisms are Lagrangian in the product — basis of…
Moment map
Equivariant map μ: M → g* for a Hamiltonian G-action, characterised by d⟨μ,X⟩ = ι_{X_M}ω for every X ∈ g. Enables symplectic reduction M//G…
Contact geometry
Odd-dimensional analogue: (M^{2n+1}, α) with α ∧ (dα)^n a volume form. Contact manifolds are boundaries of symplectic fillings; the Reeb…
Gromov non-squeezing theorem
A ball B^{2n}(r) cannot be symplectically embedded into a cylinder Z^{2n}(R) = D²(R) × ℝ^{2n–2} unless r ≤ R. Proves symplectic capacities…
Symplectic capacity
Invariant c assigning to each symplectic manifold (M,ω) a value in [0,∞] that is monotone under symplectic embedding and conformal under…
Arnold conjecture
Number of fixed points of a Hamiltonian diffeomorphism on a compact symplectic manifold is at least Σ rk H_i(M; F_p). Proved in great…
Liouville's theorem (Hamiltonian flow)
Every Hamiltonian flow X_H preserves the symplectic form ω (hence all powers ω^n, including the phase-space volume ω^n/n!). Proven by…
Symplectomorphism
A smooth map φ: (M, ω) → (M', ω') with φ*ω' = ω. The group Symp(M, ω) is infinite-dimensional in general; its linearisation at a point is…
Action-angle coordinates
Canonical coordinates (I, θ) on an integrable system in which the Hamiltonian depends only on the actions I: dθ/dt = ∂H/∂I = ω(I), dI/dt =…
Arnold-Liouville integrability theorem
On a 2n-symplectic manifold, n functionally independent Hamiltonians in pairwise Poisson involution foliate regular energy level sets by…
Kolmogorov-Arnold-Moser (KAM) theorem
For a non-degenerate integrable system H₀(I) perturbed by εH₁, invariant tori with Diophantine frequency vectors survive; the complement of…
Maslov index
Integer homotopy invariant of a loop in the Lagrangian Grassmannian Λ(n), equal to its signed intersection number with the Maslov cycle.…
Marsden-Weinstein symplectic reduction
Given a Hamiltonian G-action on (M, ω) with moment map μ: M → 𝔤*, if μ⁻¹(0) is smooth with free G-action, the quotient M//G is a symplectic…
Canonical symplectic form on T*M
Every cotangent bundle T*M has a canonical 1-form θ (Liouville/tautological form) whose exterior derivative ω = dθ is the canonical…
Poincaré–Cartan integral invariant
The 1-form λ = p dq − H dt on extended phase space has a closed 2-form dλ = ω − dH ∧ dt which is preserved along the Hamiltonian flow. …
Noether theorem (Hamiltonian form)
In the Hamiltonian formulation, a smooth observable F is conserved along the flow of the Hamiltonian vector field X_H iff its Poisson…
Legendre transform (Lagrangian ↔ Hamiltonian)
The Legendre transform L(q, q̇) ↦ H(q, p) = sup_q̇ (p·q̇ − L) is a fibre-wise convex duality which, under non-degeneracy of the Hessian…
Symplectic eigenvalue pairing
If S is a real symplectic matrix (S^T J S = J), then its spectrum is invariant under λ ↦ 1/λ; the eigenvalues therefore come in quadruples…
Williamson's normal form
Every symmetric positive-definite 2n × 2n matrix M is symplectically congruent to a diagonal form diag(D, D), where D = diag(ν₁, …, νₙ) and…
Floer homology (sketch)
Floer's chain complex CF(L₀, L₁) is generated by intersection points of two transverse Lagrangians, with differential ∂ counting…
Hofer metric on Ham(M, ω)
Hofer's bi-invariant distance on the group Ham(M, ω) of Hamiltonian diffeomorphisms is the infimum, over time-dependent Hamiltonians H_t…
Poisson manifold
A Poisson manifold is a smooth manifold M equipped with a bracket {·, ·} on C^∞(M) which is skew-symmetric, a derivation in each argument…