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That's a real ugly way of looking at it. Let's say we have a vector bundle over M with an n-dimensional fiber V. Let's also equip this vector bundle with a connection. Let's also suppose we have a smooth section of this bundle called f. Then, the covariant derivative of f with respect to this connection is a smooth linear map Δf from the tangent bundle TM to V which preserves the base point. Assume this linear map is right invertible (i.e. there exists a linear map g such that (Δf)g is the identity map) for all the fibers at the zeros of f. Then, according to the implicit function theoremIn mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables are defined as a function of the others. More precisely, that if f R m ''n rarr R n a is in R m a, the subspace of zeros of f is a submanifold. The ordinary Poisson bracketIn mathematics and classical mechanics, the Poisson bracket is a bilinear map turning two differentiable functions over a symplectic space into a function over that symplectic space. In particular, if we have two functions, A and B then : where ω is is only defined over , the space of smooth functions over M. However, using the connection, we can extend it to the space of smooth sections of f if we work with the algebra bundle with the graded algebraIn mathematics, in particular abstract algebra, a graded algebra is an algebra over a field, or more general R-algebra, in which there is a consistent notion of the weight of an element. The idea is that the weights of elements should add, when elements a of V-tensors as fibers. Assume also that under this Poisson bracket, { f, f } = 0 (note that it's not true that { g, g } = 0 in general for this "extended Poisson bracket" anymore) and { f, H } = 0 at the submanifold of zeros of f (If these brackets also happen to be zero everywhere, then we say the constraints close off shell). It turns out the right invertibility condition and the commutativity of flows conditions are independent of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.
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 constaint flows all bring the point to another point on the constrained subspace.
Since we wish to restrict ourselves to the constrained subspace only, this kind of suggests the Hamiltonian, or any other physical observableIn physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. These operations might involve submitting the system to various electromagnetic fields and ev should only be defined on that subspace. Equivalently, we can look at the equivalence classIn mathematics, given a set X and an equivalence relation ~ on X the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a :[a] { x in X | x ~ a } The notion of equivalence classes is useful for constructing s of smooth functions over the symplectic manifold which agree over the constrained subspace (the quotient algebra by the ideal generated by the f's, in other words). But the catch is, the Hamiltonian flows at the constrained subspace depends on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.
Let's look at the orbits of the constrained subspace under the action of the symplectic flow s generated by the f's. This would most definitely give a local foliation of the subspace because it satisfies integrability conditions ( 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 both agree over the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows would always lie in the same orbit at equal times. It also turns out if we have two smooth functions A1 and B1 which are constant over orbits at least over the constrained subspace (i.e. physical observables) (i.e. {A1,f}={B1,f}=0 over the constrained subspace)and another two A2 and B2 which are also constant over orbits such that A1 and B1 agrees with A2 and B2 respectively over the restrainted subspace, then their Poisson brackets {A1, B1} and {A2, B2} are also constant over orbits and agree over the constrainted subspace.
In general, we can't rule out " ergodic" flows which basically means that an orbit is dense in some open set or "subergodic" flows which basically that an orbit is dense in some submanifold of dimension greater than the orbit's dimension. Note we can't ever have self-intersecting orbits. But for most "practical" applications of first class constraints, we do not such complications and so, 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 less dimensions. In general, the quotient space is a bit "nasty" to work with when doing concrete calculations (not to mention nonlocal when working with difeomorphism constraints), so what is usually done instead is something similar. Note that the restricted submanifold is a bundle (but not a fiber bundle in general) over the quotient manifold. So, instead of working with the quotient manifold, we can work with a section of the bundle instead. This is called gauge fixing. The BIG problem is (and I really have to emphasize this, since this is a flaw in quantizing gauge theories which many physicists overlook) this bundle might not have a global section in general (This is where the "problem" of global anomalies come in, for example)!!!!! See Gribov ambiguity .
What we've just described is are irreducible first class constraints. Another complication is that Δf might not be right invertible on subspaces of the restricted submanifold of codimension 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the cotetrad formulation of general relativity at the subspace of configurations where the cotetrad field and the connection form happens to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraint s.
One way to get around this is this: For reducible constraints, we relax the condition on the right invertibility of Δf into this: Any smooth function which vanishes at the zeros of f is the fiberwise contraction of f with (a non-unique) smooth section of a -vector bundle where is the dual vector space to the constraint vector space V. (This is called the regularity condition)