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What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other '''on''' the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.
Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical [[observable]], should only be defined on that subspace. Equivalently, we can look at the [[equivalence class]] of smooth functions over the symplectic manifold, which agree on the constrained subspace (the [[quotient associative algebra|quotient algebra]] by the [[Ideal (ring theory)|ideal]] generated by the {{mvar|f}} 's, in other words).
The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.
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