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Why do we want the 2-form $\omega $ to be closed? What if it is not?

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First some terminology:

  1. A non-degenerate 2-form $\omega$ is called an almost symplectic structure.

  2. A closed 2-form $\omega$ is often called a presymplectic structure.

  3. If the 2-form $\omega$ is both non-degenerate and closed, it becomes a symplectic structure.

In the non-degenerate case, the closedness condition $$\mathrm{d}\omega~=~0\tag{C}$$ is equivalent to the Jacobi identity (JI) for the corresponding Poisson bracket (PB). In other words, conversely, a violation of the closedness condition (C) would mean a violation of the JI.

Moreover in the non-degenerate case, the closedness condition (C) (or equivalently, the JI) is the integrability condition that ensures the local existence of Darboux coordinates (aka. canonical coordinates), cf. Darboux' theorem. Conversely, the existence of Darboux coordinates in a local neighborhood $U$ implies the closedness condition (C) in that neighborhood.

For further information, see also e.g. Wikipedia$^1$; this, this, and this related SE posts; and links therein.

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$^1$ Wikipedia (August, 2015) has a concise section about motivations arising from Hamiltonian mechanics, cf. above comment by ACuriousMind. Wikipedia argues that $$\mathrm{d}H(V_H)~\equiv~\omega(V_H,V_H)~=~0\qquad\text{and}\qquad {\cal L}_{V_H}\omega~\equiv~i_{V_H}\mathrm{d}\omega~=~0 .$$ To complete Wikipedia's argument and deduce (pointwise) that $\omega$ is (i) alternating and (ii) closed, note that the Hamiltonian vector field $V_H$ needs to probe all directions in the tangent space of the point. This can be achieved by choosing the Hamiltonian generator $H$ in $2n$ different ways (because $\omega$ is non-degenerate).

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