equations for are which gives a constraint, is our Hamiltonian constraint! It could also be obtained from the Euler–Lagrange equation of motion, noting that the action depends on but not its derivative. Then the extended phase space variables , , , and are constrained to take values on this constraint-hypersurface of the extended phase space. We refer to as the `smeared' Hamiltonian constraint where is an arbitrary number. The 'smeared' Hamiltonian constraint tells us how an extended phase space variable (or function thereof) evolves with respect to : (these are actually the other Hamilton's equations). These equations describe a flow or orbit in phase space. In general we have for any phase space function . As the Hamiltonian constraint Poisson commutes with itself, it preserves itself and hence the constraint-hypersurface. The possible correlations between measurable quantities like and then correspond to `orbits' generated by the constraint within the constraint surface, each particular orbit differentiated from each other by say also measuring the value of say along with and at one -instant; after determining the particular orbit, for each measurement of we can predict the value will take. Deparametrization The other equations of Hamiltonian mechanics are Upon substitution of our action these give, These represent the fundamental equations governing our system. In the case of the parametrized clock and pendulum system we can of course recover the usual equations of motion in which is the independent variable: Now and can be deduced by We recover the usual differential equation for the simple harmonic oscillator, We also have or Our Hamiltonian constraint is then easily seen as the condition of constancy of energy! Deparametrization and the identification of a time variable with respect to which everything evolves is the opposite process of parametrization. It turns out in general that not all reparametrisation-invariant systems can be deparametrized. General relativity being a prime physical example (here the spacetime coordinates correspond to the unphysical and the Hamiltonian is a linear combination of constraints which generate spatial and time diffeomorphisms). Reason why we could deparametrize here The underlining reason why we could deparametrize (aside from the fact that we already know it was an artificial reparametrization in the first place) is the mathematical form of the constraint, namely, Substitute the Hamiltonian constraint into the original action we obtain which is the standard action for the harmonic oscillator. General relativity is an example of a physical theory