Math 2.0

Differential geometry Definition

Symplectomorphism

Builds on Symplectic manifoldDiffeomorphismDifferential formSymplectic groupHamiltonian vector field

Definition

Let (M,ω)(M, \omega) and (N,η)(N, \eta) be symplectic manifolds. A symplectomorphism from MM to NN is a diffeomorphism ϕ ⁣:M→N\phi \colon M \to N such that ϕ∗η=ω\phi^* \eta = \omega, that is ηϕ(p)(dϕp(v),dϕp(w))=ωp(v,w)\eta_{\phi(p)}\big(d\phi_p(v), d\phi_p(w)\big) = \omega_p(v, w) for all p∈Mp \in M and v,w∈TpMv, w \in T_p M.

Remarks

Example: on ℝ2\R^2 with ω=dx∧dy\omega = dx \wedge dy, let ϕ(x,y)=(x,y+f(x))\phi(x, y) = (x, y + f(x)) for a smooth function ff. It is a diffeomorphism with inverse (x,y)↦(x,y−f(x))(x, y) \mapsto (x, y - f(x)), and ϕ∗ω=dx∧(dy+f′(x) dx)=dx∧dy\phi^* \omega = dx \wedge (dy + f'(x)\, dx) = dx \wedge dy.

Non-example: ψ(x,y)=(2x,y)\psi(x, y) = (2x, y) on the same plane is a diffeomorphism, but ψ∗ω=2 dx∧dy≠ω\psi^* \omega = 2\, dx \wedge dy \ne \omega.

A linear map of (ℝ2n,ω0)(\R^{2n}, \omega_0) is a symplectomorphism exactly when it lies in the symplectic group Sp(2n,ℝ)Sp(2n, \R). The flow of a Hamiltonian vector field consists of symplectomorphisms. A symplectomorphism preserves the volume form ωn/n!\omega^n / n!, so it preserves volume; in dimension 22 the converse holds for orientation-preserving diffeomorphisms.

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Differential form, concept, no evidence levelDifferential formconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHamiltonian vector field, concept, no evidence levelHamiltonian vector fi…conceptSymplectic group, object, no evidence levelSymplectic groupobjectSymplectic manifold, concept, no evidence levelSymplectic manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptSymplectomorphism, definition, no evidence levelSymplectomorphismdefinition
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  1. Differential form concept
  2. Euclidean space object
  3. Function concept
  4. Hamiltonian vector field concept
  5. Symplectic group object
  6. Symplectic manifold concept
  7. Total derivative concept
  8. Function of several variables concept
  9. Differentiable function of several variables concept
  10. Diffeomorphism concept
  11. Symplectomorphism definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Symplectomorphism
About this page
TagQP0MY1E9RK
URIm2://QP0MY1E9RK
Kinddefinition
WritingW0 stub
MSC53D05, 53D22
DefinesSymplectomorphism
UsesSymplectic manifold, Diffeomorphism, Differential form, Symplectic group, Hamiltonian vector field
Changed2 November 2026

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