# Hamiltonian as differential manifold

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?

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.