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By the same reasoning, this constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient {{mvar|u}}<sub>2</sub>. At this point, the Hamiltonian is
:<math>
H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) ~.
</math>
<math>\vec{p}\cdot\vec{r}=0</math>,
by demanding, for consistency, that <math>\{r^2-R^2,\, H\}_{PB} = 0</math> on-shell.
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 :<math>
H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) + u_3 \vec{p}\cdot\vec{r}~,
</math>
where {{mvar|u}}<
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. The tertiary constraint's consistency condition yields
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\{\vec{p}\cdot\vec{r},\, H\}_{PB} = \frac{p^2}{m} - mgz+ \lambda r^2 -2 u_2 r^2 = 0.
</math>
:<math>
u_2 = \frac{\lambda}{2} + \frac{1}{r^2}\left(\frac{p^2}{2m}-\frac{1}{2}mgz \right).
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u_3 = -\frac{\vec{r}\cdot\vec{p}}{m r^2}~.
</math>
At this point there are ''no more constraints or consistency conditions
Putting it all together,
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The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but ''even on-shell it is nonzero''. Therefore, {{math| ''φ''<sub>2</sub>}} and {{math| ''φ''<sub>3</sub>}} are '''second class constraints''' while {{math| ''φ''<sub>1</sub>}} is a first class constraint. Note that these constraints satisfy the regularity condition.
Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. However, [[Paul Dirac|Dirac]] noticed that we can turn the underlying [[differential manifold]] of the [[symplectic manifold|symplectic space]] into a [[Poisson manifold]] using his eponymous modified bracket, called the [[Dirac bracket]], such that this ''Dirac bracket of any (smooth) function with any of the second class constraints always vanishes''. Effectively, these brackets, illustrated for this spherical surface in the [[Dirac becket ]] article, project the system back onto the constraints surface.
If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets<ref>{{Cite journal | last1 = Corrigan | first1 = E. | last2 = Zachos | first2 = C. K. | doi = 10.1016/0370-2693(79)90465-9 | title = Non-local charges for the supersymmetric σ-model | journal = Physics Letters B | volume = 88 | issue = 3–4 | pages = 273 | year = 1979 | pmid = | pmc = |bibcode = 1979PhLB...88..273C }}</ref>, ''not'' the canonical Poisson brackets to commutation relations.
Examination of the above Hamiltonian shows a number of interesting things happening. One thing to note is that, on-shell when the constraints are satisfied, the extended Hamiltonian is identical to the naive Hamiltonian, as required. Also, note that {{mvar|λ}} dropped out of the extended Hamiltonian. Since {{math| ''φ''<sub>1</sub>}} is a first class primary constraint, it should be interpreted as a generator of a gauge transformation. The gauge freedom is the freedom to choose {{mvar|λ}}, which has ceased to have any effect on the particle's dynamics. Therefore, that {{mvar|λ}} dropped out of the Hamiltonian, that {{mvar|u}}<
Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take <math>r^2-R^2</math> as a primary constraint and proceed through the formalism
===Example: Proca action===
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