Alphabeta MathHow statement and proof provenance work
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Symplectic form and symplectic manifold
Definition
A symplectic form on a smooth manifold is a smooth two-form such that
- , and
- is a symplectic vector space for every .
The pair is a symplectic manifold. Thus both closedness and pointwise nondegeneracy are required. A zero-dimensional manifold with its zero two-form satisfies the definition.
Depends on
Used by
- A nondegenerate nonclosed two-form in dimension at least four Counterexample
- Action and angle coordinates Definition
- Hamiltonian vector field and Hamiltonian function Definition
- Isotropic, coisotropic, symplectic, and Lagrangian submanifolds Definition
- Liouville vector field on an exact symplectic manifold Definition
- Symplectic and Hamiltonian Lie-group actions Definition
- Symplectic vector field Definition
- Symplectomorphisms, local symplectomorphisms, and symplectic embeddings Definition
- Backward Euler is A-stable but not symplectic Example
- Kick-then-drift symplectic Euler on the harmonic oscillator Example
- Quadratic sl2 comoment action on the symplectic plane Example
- The Moyal product for a constant Poisson tensor Example
- An infinitesimal moment map is automatically equivariant False statement
- Every nondegenerate two-form is symplectic False statement
- Every value of a moment map gives a smooth symplectic quotient False statement
- Canonical kick and drift shears satisfy the symplectic Jacobian identity Lemma
- Gaussian beams concentrated on an embedded null bicharacteristic Lemma
- The regular holonomic phase space is a symplectic cotangent bundle Lemma
- Local leapfrog and kick-then-drift symplectic Euler are symplectic Proposition
- Products and opposites of symplectic manifolds Proposition
- Exact RATTLE is symplectic on the constrained phase space Theorem
- Marsden--Weinstein--Meyer symplectic reduction Theorem
- Nondegeneracy is equivalent to a nonvanishing top wedge Theorem
Dependency tree · two levels
2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)