to illustrate constraints. The conjugate momenta are given by , , , . Note that we can't determine from the momenta. The Hamiltonian is given by . We cannot eliminate at this stage yet. We are here treating as a shorthand for a function of the symplectic space which we have yet to determine and not as an independent variable. For notational consistency, define from now on. The above Hamiltonian with the term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish, on-shell, between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, . We have the primary constraint . We require, on the grounds of consistency, that the Poisson bracket of all the constraints with the Hamiltonian vanish at the constrained subspace. In other words, the constraints must not evolve in time if they are going to be identically zero along the equations of motion. From this consistency condition, we immediately get the secondary constraint This constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient 2, enlarging the Hamiltonian to Similarly, from this secondary constraint, we find the tertiary constraint Again, one should add this constraint into the Hamiltonian, since, on-shell, no one can tell the difference. Therefore, so far, the Hamiltonian looks like where 1, 2, and 3 are still completely undetermined. Note that, frequently, all constraints that are found from consistency conditions are referred to as secondary constraints and secondary, tertiary, quaternary, etc., constraints are not distinguished. We keep turning the crank, demanding this new constraint have vanishing Poisson bracket We might despair and think that there is no end to this, but because one of the new Lagrange multipliers has shown up, this is not a new constraint, but a condition that fixes the Lagrange multiplier: Plugging this into our Hamiltonian gives us (after a little algebra) Now that there are new terms in the Hamiltonian, one should go back and check the consistency conditions for the primary and secondary constraints. The secondary constraint's consistency condition gives Again, this is not a new constraint; it only determines that At this point there are no more constraints or consistency conditions to check! Putting it all together, . When finding the equations of motion, one should use the above Hamiltonian, and as long as one is careful to never use constraints before taking derivatives