A basic ingredient of the canonical formalism is the Hamiltonian function. In the case of a regular theory (that is, with a non-singular Hessian matrix) it defines, by use of the Poisson bracket, the vector field that generates the time evolution – the dynamics – in phase space. The Hamiltonian is given in that case as the projection to phase space of the Lagrangian energy E = ∂L/∂q ̇ − L.
This procedure to define the Hamiltonian will still work in the singular case if the energy satisfies the conditions of projectability (7). Indeed we can readily check that ΓμE = 0, so we have a canonical Hamiltonian Hc, defined as a function on phase space whose pullback is the Lagrangian energy, FL ∗ Hc = E. It was Dirac that first realized in the general setting of constrained systems that a Hamiltonian always existed.
There is a slight difference, though, from the regular case, for now there is an ambiguity in the definition of Hc. In fact, since FL ∗ φμ = 0, many candidates for canonical Hamiltonians are available, once we are given one. In fact, Hc + vμφμ — with vμ(q, q ̇; t) arbitrary functions and with summation convention for μ – is as good as Hc as a canonical Hamiltonian. This “slight difference” is bound to have profound consequences: it is the door to gauge freedom.