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==Constrained Hamiltonian dynamics from a Lagrangian gauge theory==
First of all, we will assume the [[action (physics)|action]] is the integral of a local [[Lagrangian (field theory)|Lagrangian]] that only depends up to the first derivative of the fields. The analysis of more general cases, while possible is more complicated. When going over to the Hamiltonian formalism, we find there are constraints. Recall that in the action formalism, there are [[on shell]] and [[off shell]] configurations. The constraints that hold off shell are called primary constraints while those that only hold on shell are called secondary constraints.
==Examples==
Look at the dynamics of a single point particle of mass ''m'' with no internal degrees of freedom moving in a [[pseudo-Riemannian]] spacetime manifold ''S'' with [[metric tensor|metric]] '''g'''. Assume also that the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon [[Parametric curve#Reparametrization and equivalence relation|reparametrization invariance]]). Then, its [[symplectic manifold|symplectic space]] is the [[cotangent bundle]] T*S with the canonical symplectic form ω. If we coordinatize ''T'' * ''S'' by its position ''x'' in the base manifold ''S'' and its position within the cotangent space '''p''', then we have a constraint
:''f'' = ''m''<sup>2</sup> −'''g'''(''x'')<sup>−1</sup>('''p''','''p''') = 0.
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:<math>H=\vec{p}\cdot\dot{\vec{r}}+p_\lambda \dot{\lambda}-L=\frac{p^2}{2m}+p_\lambda \dot{\lambda}+mgz-\frac{\lambda}{2}(r^2-R^2)</math>.
We cannot eliminate <math>\dot{\lambda}</math> at this stage yet. We are here treating <math>\dot{\lambda}</math> as a shorthand for a function of the [[symplectic manifold|symplectic space]] which we have yet to determine and ''not'' an independent variable. For notational consistency, define <math>u_1=\dot{\lambda}</math> from now on. The above Hamiltonian with the {{math|''p''<sub>''λ''</sub>}} 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, <math>\dot{\lambda}=u_1</math>.
We have the [[primary constraint]]
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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>\phi_2</math> and <math>\phi_3</math> are '''second class constraints''' while <math>\phi_1</math> 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. But [[Paul Dirac|Dirac]] noticed that we can turn the underlying [[differential manifold]] of the [[symplectic manifold|symplectic space]] into a [[Poisson manifold]] using a different bracket, called the [[Dirac bracket]], such that the Dirac bracket of any (smooth) function with any of the second class constraints always vanishes and a couple of other nice properties.
If one wanted to canonically quantize this system, then, one needs to promote the canonical Dirac brackets<ref>{{cite doi|10.1016/0370-2693(79)90465-9|noedit}}</ref> not the canonical Poisson brackets to commutation relations.
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