Lagrangian maps and are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form. Symbolically: where τ∗ω2 denotes the pull back of ω2 by τ. Special cases and generalizations A symplectic manifold is exact if the symplectic form is exact. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold. The canonical symplectic form is exact. A symplectic manifold endowed with a metric that is compatible with the symplectic form is an almost Kähler manifold in the sense that the tangent bundle has an almost complex structure, but this need not be integrable. Symplectic manifolds are special cases of a Poisson manifold. A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued -form; it is utilized in Hamiltonian field theory. See also —an odd-dimensional counterpart of the symplectic manifold. Citations General and cited references Further reading Differential topology Hamiltonian mechanics Smooth manifolds Symplectic geometry