The Hamiltonian constraint arises from any theory that admits a Hamiltonian formulation and is reparametrisation-invariant. The Hamiltonian constraint of general relativity is an important non-trivial example. In the context of general relativity, the Hamiltonian constraint technically refers to a linear combination of spatial and time diffeomorphism constraints reflecting the reparametrizability of the theory under both spatial as well as time coordinates. However, most of the time the term Hamiltonian constraint is reserved for the constraint that generates time diffeomorphisms. Simplest example: the parametrized clock and pendulum system Parametrization In its usual presentation, classical mechanics appears to give time a special role as an independent variable. This is unnecessary, however. Mechanics can be formulated to treat the time variable on the same footing as the other variables in an extended phase space, by parameterizing the temporal variable(s) in terms of a common, albeit unspecified parameter variable. Phase space variables being on the same footing. Say our system comprised a pendulum executing a simple harmonic motion and a clock. Whereas the system could be described classically by a position x=x(t), with x defined as a function of time, it is also possible to describe the same system as x() and t() where the relation between x and t is not directly specified. Instead, x and t are determined by the parameter , which is simply a parameter of the system, possibly having no objective meaning in its own right. The system would be described by the position of a pendulum from the center, denoted , and the reading on the clock, denoted . We put these variables on the same footing by introducing a fictitious parameter , whose 'evolution' with respect to takes us continuously through every possible correlation between the displacement and reading on the clock. Obviously the variable can be replaced by any monotonic function, . This is what makes the system reparametrisation-invariant. Note that by this reparametrisation-invariance the theory cannot predict the value of or for a given value of but only the relationship between these quantities. Dynamics is then determined by this relationship. Dynamics of this reparametrization-invariant system The action for the parametrized Harmonic oscillator is then where and are canonical coordinates and and are their conjugate momenta respectively and represent our extended phase space (we will show that we can recover the usual Newton's equations from this expression). Writing the action as we identify the as Hamilton's