# Why should a symplectic form be closed?

Thanks for reading my question. I'm wonder why a symplectic form should be closed. I found many different answers in the internet, but it sounds like a technical requirement (if we omit this requisit, we obtain almost symplectic structures, insteresting as well). Why do yo think? I just want to have a fresh perspective. Thank you in advance.

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This has been discussed on MathsOverflow. It seems that things such as conservation of energy and momentum would not be true if the two form were not closed. mathoverflow.net/questions/19932/… –  Fly by Night Jan 29 '13 at 22:01
Thank you very much! I'll check that post. –  math_failure Jan 31 '13 at 16:42

• A closed $2$-form represents a cohomology class in $H^2(M; \Bbb R)$.
• When $\omega$ is closed, we get a one-to-one correspondence between $1$-parameter groups of symplectomorphisms and symplectic vector fields. We also get Hamiltonian diffeomorphisms from this, which turn out to be interesting for symplectic geometers.