Symplectic form

See Hamiltonian mechanics#Symplectic form.
It is a closed non-degenerate differential 2-form. It defines a symplectic manifold. At the linear level it pairs the coordinates of the space: see symplectic vector space.

In finite dimensional manifolds we can use, I think, the Pfaff-Darboux theorem to obtain coordinates which provide a particularly simple expression for the symplectic form. According to @olver86 page 390 it is not true in the infinite dimensional case.

Discussion

In any cotangent bundle $T^{*} Q$ we have a canonical 1-form known as tautological 1-form, Poincaré 1-form, Liouville 1-form, symplectic potential or canonical 1-form. Let's see how is defined. We are looking for an element $λ \in Ω^{1} (T^{*} Q)$ , so consider an element (in any local chart, it doesn't have to be the special one provide by the Lagrangian) $(q, p) \in T^{*} Q .$ We want a definition for a map

λ_{(q, p)} : T_{(q, p)} (T^{*} Q) ⟼ R

Since in local chart we see elements in the form

(q, p, v, η) \in T_{(q, p)} (T^{*} Q)

it seems natural to forget $η$ and take

λ_{(q, p)} (q, p, v, η) = p (v)

The point is that we can define it without coordinates!
Consider

π : T^{*} Q ⟼ Q

and

d π : T (T^{*} Q) ⟼ T Q .

We can define

λ : T^{*} Q ⟼ T^{*} (T^{*} Q)

λ_{α} = π^{*} (α) .

Incidentally (and surprisingly), the tautological 1-form $λ$ let us define the action!! See Hamiltonian mechanics#Variational principle for Hamiltonian mechanics.

Now, we can take the exterior derivative, $ω := d λ$ . This is called the Poincaré 2-form, or symplectic 2-form. It is canonical for every cotangent bundle and verifies:

It is closed.
It is nondegenerate. That is, for every vector $X \in T (T^{*} Q)$ , the 1-form $i_{X} ω = ω (X, -)$ is not null.

Any manifold equipped with a 2-form satisfying both conditions is called a symplectic manifold.