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{{distinguish|Primary constraint}}
{{broader|Dirac bracket}}
In [[physics]], a '''first-class constraint''' is a dynamical quantity in a constrained [[Hamiltonian mechanics|Hamiltonian]] system whose [[Poisson bracket]] with all the other constraints vanishes on the '''constraint surface''' in [[phase space]] (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first-class constraint, one assumes that there are no '''second-class constraints''', or that they have been calculated previously, and their [[Dirac bracket]]s generated.<ref name=FysikSuSePDF>{{cite web |author1=Ingemar Bengtsson |title=Constrained Hamiltonian Systems |url=http://3dhouse.se/ingemar/Nr13.pdf |publisher=Stockholm University |access-date=29 May 2018 |quote=We start from a Lagrangian <math>L(q, \dot q),</math> derive the canonical momenta, postulate the naive Poisson brackets, and compute the Hamiltonian. For simplicity, one assumes that no second class constraints occur, or if they do, that they have been dealt with already and the naive brackets replaced with Dirac brackets. There remain a set of constraints [...]}}</ref>
First- and second
The terminology of first- and second
==Poisson brackets==
Suppose we have some constraints
:<math> f_i(x)=0, </math>
for ''n'' smooth functions
:<math>\{ f_i \}_{i= 1}^n</math>
These will only be defined [[chart (topology)|chartwise]] in general. Suppose that everywhere on the constrained set, the ''n'' derivatives of the ''n'' functions are all [[linearly independent]] and also that the [[Poisson bracket]]s
:<math>\{f_i,f_j\}</math>
and
:<math>\{f_i,H\}</math>
all vanish on the constrained subspace.
:<math>\{f_i,f_j\}=\sum_k c_{ij}^k f_k</math>
for some smooth functions <math>c_{ij}^k</math> — there is a theorem showing this; and
:<math>\{f_i,H\}=\sum_j v_i^j f_j</math>
for some smooth functions <math>v_i^j</math>.
This can be done globally, using a [[partition of unity]]. Then, we say we have an irreducible '''first-class constraint''' (''irreducible'' here is in a different sense from that used in [[representation theory]]).
==Geometric theory==
For a more elegant way, suppose given a [[vector bundle]] over <math>\mathcal M</math>, with
Then the [[covariant derivative]] of {{mvar|f}} with respect to the connection is a smooth [[linear map]] <math>\nabla f</math> from the [[tangent bundle]] <math>T\mathcal M</math> to <math>V</math>, which preserves the [[base point]]. Assume this linear map is right [[invertible]] (i.e. there exists a linear map <math>g</math> such that <math>(\Delta f)g</math> is the [[identity function|identity map]]) for all the fibers at the zeros of {{mvar|f}}. Then, according to the [[implicit function theorem]], the subspace of zeros of {{mvar|f}} is a [[submanifold]].
The ordinary [[Poisson bracket]] is only defined over <math>C^{\infty}(M)</math>, the space of smooth functions over ''M''. However, using the connection, we can extend it to the space of smooth sections of {{mvar|f}} if we work with the [[algebra bundle]] with the [[graded algebra]] of ''V''-tensors as fibers.
Assume also that under this Poisson bracket, <math>\{f,f\}=0</math> (note that it's not true that <math>\{g,g\}=0</math> in general for this "extended Poisson bracket" anymore) and <math>\{f,H\}=0</math> on the submanifold of zeros of
==Intuitive meaning==
What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other '''on''' the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.
Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical [[observable]], should only be defined on that subspace. Equivalently, we can look at the [[equivalence class]] of smooth functions over the symplectic manifold, which agree on the constrained subspace (the [[quotient associative algebra|quotient algebra]] by the [[Ideal (ring theory)|ideal]]
The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.
Look at the [[orbit (group theory)|orbits]] of the constrained subspace under the action of the [[Symplectomorphism|symplectic
In general, one cannot rule out "[[ergodic]]" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have [[self-intersecting]] orbits.
For most "practical" applications of first-class constraints, we do not see such complications: the [[Quotient space (topology)|quotient space]] of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a [[differentiable manifold]], which can be turned into a [[symplectic manifold]] by projecting the [[symplectic form]] of M onto it (this can be shown to be [[well defined]]). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.
In general, the quotient space is a bit
The ''major'' problem is this bundle might not have a [[global section]] in general. This is where the "problem" of [[global anomaly|global anomalies]] comes in, for example.
What have been described are irreducible first-class constraints. Another complication is that Δf might not be [[right invertible]] on subspaces of the restricted submanifold of [[codimension]] 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the [[cotetrad]] formulation of [[general relativity]], at the subspace of configurations where the [[cotetrad field]] and the [[connection form]] happen to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraints.
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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==
If we coordinatize ''T'' * ''S'' by its position {{mvar|x}} in the base manifold {{mvar|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.
The Hamiltonian
Consider now the case of a [[
:{{math|''ρ(l)[σ]''}}
as
:{{math|''l[σ]''}}
for simplicity. Let '''A''' be the {{mvar|L}}-valued [[connection form]] of the theory. Note that the '''A''' here differs from the '''A''' used by physicists by a factor of {{mvar|i}} and {{mvar|g}}. This agrees with the mathematician's convention.
The action {{mvar|S}} is given by
:<math>S[\mathbf{A},\sigma]=\int d^dx \frac{1}{4g^2}\eta((\mathbf{g}^{-1}\otimes \mathbf{g}^{-1})(\mathbf{F},\mathbf{F}))+\frac{1}{2}\alpha(\mathbf{g}^{-1}(D\sigma,D\sigma))</math>
where '''g''' is the Minkowski metric, '''F''' is the [[curvature form]]
:<math>d\
(no {{mvar|i}}s or {{mvar|g}}s!) where the second term is a formal shorthand for pretending the Lie bracket is a commutator, {{mvar|D}} is the covariant derivative
:Dσ = dσ − '''A'''[σ]
and {{mvar|α}} is the orthogonal form for {{mvar|ρ}}.
<!--I hope I have all the signs and factors right. I can't guarantee it.-->
What is the Hamiltonian version of this model? Well, first, we have to split '''A''' noncovariantly into a time component {{mvar|φ}} and a spatial part {{vec|''A''}}. Then, the resulting symplectic space has the conjugate variables {{mvar|σ}}, {{math|''π<sub>σ</sub>''}} (taking values in the underlying vector space of <math>\bar{\rho}</math>, the dual rep of {{mvar|ρ}}), {{vec|''A''}}, {{vec|''π''}}<sub>''A''</sub>, ''φ'' and ''π<sub>φ</sub>''. For each spatial point, we have the constraints, ''π<sub>φ</sub>''=0 and the [[Gaussian constraint]]
:<math>\vec{D}\cdot\vec{\pi}_A-\rho'(\pi_\sigma,\sigma)=0</math>
where since {{mvar|ρ}} is an [[intertwiner]]
:<math>\rho:L\otimes V\rightarrow V</math>,
{{mvar|ρ}} ' is the dualized intertwiner
:<math>\rho':\bar{V}\otimes V\rightarrow L</math>
({{mvar|L}} is self-dual via {{mvar|η}}). The Hamiltonian,
:<math>H_f=\int d^{d-1}x \frac{1}{2}\alpha^{-1}(\pi_\sigma,\pi_\sigma)+\frac{1}{2}\alpha(\vec{D}\sigma\cdot\vec{D}\sigma)-\frac{g^2}{2}\eta(\vec{\pi}_A,\vec{\pi}_A)-\frac{1}{2g^2}\eta(\mathbf{B}\cdot \mathbf{B})-\eta(\pi_\phi,f)-<\pi_\sigma,\phi[\sigma]>-\eta(\phi,\vec{D}\cdot\vec{\pi}_A).</math>
The last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by {{mvar|f}}. In fact, since the last three terms vanish for the constrained states, we may drop them.
==Second-class constraints==
In a constrained Hamiltonian system, a dynamical quantity is '''second-class''' if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a '''second-class constraint'''.
See [[Dirac bracket]]s for diverse illustrations.
===An example: a particle confined to a sphere===
Before going on to the general theory,
In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For
The action is given by
:<math>S=\int dt L=\int dt \left[\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2)-mgz+\frac{\lambda}{2}(x^2+y^2+z^2-R^2)\right]</math>
where the last term is the [[Lagrange multiplier]] term enforcing the constraint.
Of course, as indicated, we could have just used different, non-redundant, spherical [[coordinates]] and written it as
:<math>S=\int dt \left[\frac{mR^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mgR\cos(\theta)\right]</math>
instead, without extra constraints; but we are considering the former coordinatization to illustrate constraints.
The [[conjugate momentum|conjugate momenta]] are given by
:<math>p_x=m\dot{x}</math>, <math>p_y=m\dot{y}</math>, <math>p_z=m\dot{z}</math>, <math>p_\lambda=0</math>.
Note that we can't determine {{math|{{overset|•|''λ''}}}} from the momenta.
The [[Hamiltonian mechanics|Hamiltonian]] is given by
:<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 {{overset|•|''λ''}} at this stage yet. We are here treating {{overset|•|''λ''}} as a shorthand for a function of the [[symplectic manifold|symplectic space]] which we have yet to determine and ''not'' as an independent variable. For notational consistency, define {{math| ''u''<sub>1</sub> {{=}} {{overset|•|''λ''}} }} 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| {{overset|•|''λ''}} {{=}} ''u''<sub>1</sub>}}.
We have the [[primary constraint]]
:{{math|''p<sub>λ</sub>''{{=}}0}}.
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 [[
<math>\begin{align}
0&=\{H,p_\lambda\}_\text{PB}\\
&=\sum_{i}\frac{\partial H}{\partial q_i}\frac{\partial p_\lambda}{\partial p_i}-\frac{\partial H}{\partial p_i}\frac{\partial p_\lambda}{\partial q_i}\\
&=\frac{\partial H}{\partial \lambda}\\
&=\frac{1}{2}(r^2-R^2)\\
&\Downarrow\\
0&=r^2-R^2
\end{align}</math>
This constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient {{mvar|u}}<sub>2,</sub> enlarging the Hamiltonian to
:<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>\
0&=\{H,r^2-R^2\}_{PB}\\
&=\{H,x^2\}_{PB}+\{H,y^2\}_{PB}+\{H,z^2\}_{PB}\\
&=\frac{\partial H}{\partial p_x}2x+\frac{\partial H}{\partial p_y}2y+\frac{\partial H}{\partial p_z}2z\\
&=\frac{2}{m}(p_xx+p_yy+p_zz)\\
&\Downarrow\\
0&=\vec p\cdot\vec r
\end{align}</math>
:<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 We keep turning the crank, demanding this new constraint have vanishing [[Poisson bracket]]
:<math>
0=\{\vec{p}\cdot\vec{r},\, H\}_{PB} = \frac{p^2}{m} - mgz+ \lambda r^2 -2 u_2 r^2
</math>
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:
:<math>
u_2 = \frac{\lambda}{2} + \frac{1}{r^2}\left(\frac{p^2}{2m}-\frac{1}{2}mgz \right).
</math>
Plugging this into our Hamiltonian gives us (after a little algebra)
<math>
H = \frac{p^2}{2m}(2-\frac{R^2}{r^2}) + \frac{1}{2}mgz(1+\frac{R^2}{r^2})+u_1p_\lambda+u_3\vec p \cdot\vec r
</math>
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
:<math>
\frac{2}{m}\vec{r}\cdot\vec{p} + 2 u_3 r^2 = 0.
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Again, this is ''not'' a new constraint; it only determines that
:<math>
u_3 = -\frac{\vec{r}\cdot\vec{p}}{m r^2}~.
</math>
At this point there are ''no more constraints or consistency conditions to check''!
Putting it all together,
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</math>
Before analyzing the Hamiltonian, consider the three constraints
:<math>
\
</math>
:<math>
\{\
</math>
The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but ''even on-shell it is nonzero''. Therefore,
Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace.
Effectively, these brackets (illustrated for this spherical surface in the [[Dirac bracket]] 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 |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
Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take
{{see also|Dirac bracket}}
===Example: Proca action===
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and
:<math>B_{ij} \equiv \frac{\partial A_j}{\partial x_i} - \frac{\partial A_i}{\partial x_j}</math>.
<math>(\vec{A},-\vec{E})</math> and <math>(\phi,\pi)</math> are [[canonical variables]]. The second
:<math>\pi \approx 0</math>
and
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:<math>H = \int d^dx \left[ \frac{1}{2}E^2 + \frac{1}{4}B_{ij}B_{ij} - \pi\nabla\cdot\vec{A} + \vec{E}\cdot\nabla\phi + \frac{m^2}{2}A^2 - \frac{m^2}{2}\phi^2\right]</math>.
==See also==
*[[Dirac bracket]]
*[[Holonomic constraint]]
*[[Analysis of flows]]
==References==
{{reflist}}
==Further reading==
* {{Cite journal | last1 = Falck | first1 = N. K. | last2 = Hirshfeld | first2 = A. C. | doi = 10.1088/0143-0807/4/1/003 | title = Dirac-bracket quantisation of a constrained nonlinear system: The rigid rotator | journal = European Journal of Physics | volume = 4 | pages = 5–9 | year = 1983 | issue = 1 |bibcode = 1983EJPh....4....5F | s2cid = 250845310 }}
* {{Cite journal | last1 = Homma | first1 = T. | last2 = Inamoto | first2 = T. | last3 = Miyazaki | first3 = T. | doi = 10.1103/PhysRevD.42.2049 | title = Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space | journal = Physical Review D | volume = 42 | issue = 6 | pages = 2049–2056 | year = 1990 | pmid = 10013054 |bibcode = 1990PhRvD..42.2049H }}
[[Category:Classical mechanics]]
[[Category:Theoretical physics]]
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