Mathematics.

symplectic geometry

Symplectic Geometry

Differential Geometry95 minDifficulty9 out of 10

You should know: differential forms, smooth manifolds

Overview

Symplectic geometry is the study of smooth manifolds equipped with a closed, non-degenerate 2-form ω (a symplectic form). It is the natural geometric framework for classical Hamiltonian mechanics: the phase space of a physical system is a symplectic manifold, Hamilton's equations describe the flow of a vector field preserving ω, and conserved quantities correspond to symmetries via Noether's theorem. Unlike Riemannian geometry, symplectic geometry has no local invariants (Darboux's theorem: all symplectic manifolds of the same dimension are locally equivalent) — all the interesting geometry is global.

Intuition

In classical mechanics, the state of a system is described by positions qⁱ and momenta pᵢ. The symplectic form ω = Σ dqⁱ ∧ dpᵢ encodes the fundamental Poisson bracket structure {qⁱ, pⱼ} = δⁱⱼ. Hamilton's equations can be written as: the flow of the Hamiltonian H is the unique vector field X_H satisfying ι_{X_H}ω = −dH. Energy conservation follows from ω(X_H, X_H) = 0. Unlike Riemannian geometry, symplectic geometry lacks a local notion of distance — its invariants are purely topological (such as symplectic capacities).

Formal Definition

Definition

A symplectic manifold is a smooth manifold M of even dimension 2n equipped with a symplectic form ω — a differential 2-form that is both closed and non-degenerate.

dω=0d\omega = 0
Closed
ωn=ω∧⋯∧ω≠0\omega^n = \omega \wedge \cdots \wedge \omega \neq 0
Non-degenerate (ω^n is a volume form)
ω=∑i=1ndqi∧dpi\omega = \sum_{i=1}^n dq^i \wedge dp_i
Standard symplectic form on ℝ²ⁿ with coordinates (q¹,...,qⁿ,p₁,...,pₙ)
ιXHω=−dH,q˙i=∂H∂pi,  p˙i=−∂H∂qi\iota_{X_H}\omega = -dH,\quad \dot{q}^i = \frac{\partial H}{\partial p_i},\; \dot{p}_i = -\frac{\partial H}{\partial q^i}
Hamilton's equations via the Hamiltonian vector field X_H
{f,g}=ω(Xf,Xg)=∑i(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi)\{f, g\} = \omega(X_f, X_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)
Poisson bracket

Properties

Symplectic manifolds are even-dimensional

dim⁡M=2n (non-degeneracy of ω forces even dimension)\dim M = 2n \text{ (non-degeneracy of } \omega \text{ forces even dimension)}

Symplectomorphisms preserve symplectic structure

f:(M,ω)→(N,σ) is a symplectomorphism if f∗σ=ωf: (M, \omega) \to (N, \sigma) \text{ is a symplectomorphism if } f^*\sigma = \omega

Lagrangian submanifolds

L⊆(M,ω) is Lagrangian if dim⁡L=n=12dim⁡M and ω∣L=0L \subseteq (M, \omega) \text{ is Lagrangian if } \dim L = n = \frac{1}{2}\dim M \text{ and } \omega|_L = 0

Theorems

Theorem 1: Darboux's Theorem
Every symplectic manifold (M,ω) of dimension 2n is locally symplectomorphic to (R2n,∑dqi∧dpi).\text{Every symplectic manifold } (M, \omega) \text{ of dimension } 2n \text{ is locally symplectomorphic to } (\mathbb{R}^{2n}, \sum dq^i \wedge dp_i).
Theorem 2: Liouville's Theorem
The Hamiltonian flow preserves the symplectic form: ϕt∗ω=ω, and hence preserves the volume form ωn.\text{The Hamiltonian flow preserves the symplectic form: } \phi_t^* \omega = \omega, \text{ and hence preserves the volume form } \omega^n.
Theorem 3: Gromov's Non-Squeezing Theorem
A ball B2n(r) cannot be symplectically embedded into B2(R)×R2n−2 if r>R.\text{A ball } B^{2n}(r) \text{ cannot be symplectically embedded into } B^2(R) \times \mathbb{R}^{2n-2} \text{ if } r > R.

Worked Examples

  1. 1

    Check closedness: ω = Σ dqⁱ ∧ dpᵢ is already written as a sum of exact 2-forms.

    dω=∑id(dqi∧dpi)=∑id2qi⏟0∧dpi−dqi∧d2pi⏟0=0d\omega = \sum_i d(dq^i \wedge dp_i) = \sum_i \underbrace{d^2q^i}_{0} \wedge dp_i - dq^i \wedge \underbrace{d^2p_i}_{0} = 0
  2. 2

    Check non-degeneracy: ωⁿ = (Σ dqⁱ ∧ dpᵢ)ⁿ = n! dq¹ ∧ dp₁ ∧ ... ∧ dqⁿ ∧ dpₙ ≠ 0.

    ωn=n! dq1∧dp1∧⋯∧dqn∧dpn≠0\omega^n = n!\, dq^1 \wedge dp_1 \wedge \cdots \wedge dq^n \wedge dp_n \neq 0
  3. 3

    Both conditions are satisfied, so (T*ℝⁿ, ω) is symplectic.

✓ Answer

T*ℝⁿ with ω = Σ dqⁱ ∧ dpᵢ is symplectic: ω is closed (dω = 0) and non-degenerate (ωⁿ ≠ 0).

Practice Problems

Mediumfree response

Explain why a symplectic manifold must be even-dimensional.

Hardproof writing

Prove that Hamiltonian flow preserves the symplectic form (Liouville's theorem).

Common Mistakes

Common Mistake

Symplectic and Riemannian geometry are equivalent because both use 2-tensors

A Riemannian metric is a symmetric positive-definite (0,2) tensor; a symplectic form is antisymmetric and non-degenerate. Riemannian geometry has rich local invariants (curvature); symplectic geometry has none locally (Darboux).

Common Mistake

Every even-dimensional manifold is symplectic

Not every even-dimensional manifold admits a symplectic structure. The sphere S² is symplectic (area form), but S⁴ is not (Gromov's non-squeezing and characteristic class obstructions).

Quiz

Darboux's theorem states that locally every symplectic manifold is:
The Hamiltonian vector field X_H is defined by:
A Lagrangian submanifold L of (M²ⁿ, ω) satisfies:

Historical Background

Symplectic geometry grew from the Hamiltonian formulation of classical mechanics developed by Hamilton and Jacobi in the 1830s–1840s. The word 'symplectic' was introduced by Hermann Weyl in 1939 (from Greek for 'complex', replacing 'complex' to avoid confusion). The geometric framework was systematised by Sophus Lie, Darboux, and Poincaré. In the twentieth century, symplectic geometry flourished through Moser's 1965 stability theorem, Arnold's work on integrable systems, and Gromov's 1985 introduction of J-holomorphic curves (which founded symplectic topology). McDuff and Salamon's textbook (1995) is the standard modern reference.

  1. 1833

    Hamilton introduces the Hamiltonian formulation of mechanics, identifying phase space structure

    William Rowan Hamilton

  2. 1882

    Darboux proves local normal form theorem for symplectic structures

    Jean Gaston Darboux

  3. 1939

    Weyl coins the term 'symplectic' in The Classical Groups

    Hermann Weyl

  4. 1985

    Gromov introduces J-holomorphic curves, launching symplectic topology

    Mikhail Gromov

Summary

  • A symplectic manifold (M, ω) has a closed (dω=0) and non-degenerate (ωⁿ≠0) 2-form ω; its dimension is always even.
  • Darboux's theorem: all symplectic manifolds of the same dimension are locally equivalent — there are no local symplectic invariants.
  • Hamiltonian vector fields X_H are defined by ι_{X_H}ω = −dH and generate Hamiltonian flows preserving ω.
  • The Poisson bracket {f,g} = ω(X_f, X_g) makes C∞(M) a Lie algebra, encoding the structure of classical mechanics.
  • Global symplectic invariants (symplectic capacities, Gromov's non-squeezing) are studied in symplectic topology.

References

  1. BookMcDuff, D. and Salamon, D. — Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017
  2. BookArnold, V. I. — Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989
  3. WebsiteWikipedia — Symplectic geometry
  4. WebsitenLab — symplectic geometry