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Look at the [[orbit (group theory)|orbits]] of the constrained subspace under the action of the [[symplectic flow]]s generated by the ''f'''s. This gives a local [[foliation]] of the subspace because it satisfies [[integrability condition]]s ([[Frobenius theorem (differential topology)|Frobenius theorem]]). It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times. It also turns out if we have two smooth functions ''A''<sub>1</sub> and ''B''<sub>1</sub>, which are constant over orbits at least on the constrained subspace (i.e. physical observables) (i.e. {A<sub>1</sub>,f}={B<sub>1</sub>,f}=0 over the constrained subspace)and another two A<sub>2</sub> and B<sub>2</sub>, which are also constant over orbits such that A<sub>1</sub> and B<sub>1</sub> agrees with A<sub>2</sub> and B<sub>2</sub> respectively over the restrained subspace, then their Poisson brackets {A<sub>1</sub>, B<sub>1</sub>} and {A<sub>2</sub>, B<sub>2</sub>} are also constant over orbits and agree over the constrained subspace.
In general,
For most "practical" applications of first-class constraints, we do not see such complications: the [[quotient space]] of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a [[differentiable manifold]], which can be turned into a [[symplectic manifold]] by projecting the [[symplectic form]] of M onto it (this can be shown to be [[well defined]]). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.
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