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I know that phase space $(q^i, p^i)$ can be treated as manifold. But for me defining hamiltonian as a function also leads to new manifold $(q^i, p^i, H(q^i,p^i))$. Like map from open set $(q^i, p^i) \mapsto (q^i, p^i, H(q^i,p^i))$. Am I wrong about this?

I'm asking this question cause I have troubles with symplectic form. Do we define it only on phase space or we need hamiltonian too?

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The hamiltonian is an $\mathbf{R}$-valued function on $2n$ dimensional phase space $\mathcal{M}$. The $2n+1$ dimensional space you suggest is basically the graph (in the high school algebra sense) of the hamiltonian function on $\mathcal{M}$. The symplectic form or Poisson bracket is defined on $\mathcal{M}$, not on the larger space.

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