The canonical treatment of the symmetry reductions of GR requires the understanding of constrained
Hamiltonian systems. In the cases that we are going to discuss (and leaving aside functional analytic issues
relevant for field theories [101]), the starting point consists of the classical unconstrained configuration
space
of the model and the cotangent bundle
over
endowed with a suitable symplectic form
. A dynamical Hamiltonian system is said to be constrained if the physical states are restricted to
belonging to a submanifold
of the phase space
, and the dynamics are such that time evolution
takes place within
[101]. In the examples relevant for us the space
will be globally defined by the
vanishing of certain sufficiently regular constraint functions,
. In the case of GR these
constraint functions are the integrated version of the scalar and vector constraints and the
subindex
refers to lapse and shift choices (see, for example, [8]). Notice, however, that there
exist infinitely-many constraint equations that define the same submanifold
. The choice of
one representation or another is, in practice, dictated by the variables used to describe the
physical system. We will assume that
is a first-class submanifold of
. This is geometrical
property that can be expressed in terms of the concrete constraint equations describing
as
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Living Rev. Relativity 13, (2010), 6
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