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cianfa72

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- About the definition of symplectic manifold structure employed in the hamiltonian formulation of classical mechanics

Hi, in the Hamiltonian formulation of classical mechanics, the phase space is a symplectic manifold. Namely there is a closed non-degenerate 2-form ##\omega## that assign a symplectic structure to the ##2m## even dimensional manifold (the phase space).

As explained here Darboux's theorem since ##\omega## is by definition closed from Poincare lemma there exist locally a 1-form ##\theta## such that locally ##\omega = d\theta##.

However I've not a clear understanding why such ##d\theta## fulfills the Darboux's theorem hypothesis hence there are local canonical coordinates such that ##\omega## can be written as

$$\omega = dq_i \wedge dp_i$$

If ##\omega## was a rank ##m## form then by definition ##(d\theta)^m \neq 0## and of course ##\theta \wedge (d\theta)^m = 0## since it would be a ##2m+1## form defined on a 2m-dimensional manifold.

So the question is: why ##\omega## is assumed to be a 2-form with constant rank ##m## ? Thanks.

As explained here Darboux's theorem since ##\omega## is by definition closed from Poincare lemma there exist locally a 1-form ##\theta## such that locally ##\omega = d\theta##.

However I've not a clear understanding why such ##d\theta## fulfills the Darboux's theorem hypothesis hence there are local canonical coordinates such that ##\omega## can be written as

$$\omega = dq_i \wedge dp_i$$

If ##\omega## was a rank ##m## form then by definition ##(d\theta)^m \neq 0## and of course ##\theta \wedge (d\theta)^m = 0## since it would be a ##2m+1## form defined on a 2m-dimensional manifold.

So the question is: why ##\omega## is assumed to be a 2-form with constant rank ##m## ? Thanks.

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