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{{distinguish|Primary constraint}}
{{
First- and second
The terminology of first- and second
==Poisson brackets==
Consider a [[
Suppose we have some constraints
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This means we can write
:<math>\{f_i,f_j\}=\sum_k c_{ij}^k f_k</math>
for some smooth functions
:<math>\{f_i,H\}=\sum_j v_i^j f_j</math>
for some smooth functions
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 <math>n</math>-dimensional [[Fiber (mathematics)|fiber]] <math>V</math>. Equip this vector bundle with a [[connection form|connection]]. Suppose too we have a [[Section (fiber bundle)|smooth section]] {{mvar|f}} of this bundle.
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]].
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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 {{mvar|f}} (If these brackets also happen to be zero everywhere, then we say the constraints close [[off shell]]). It turns out the right invertibility condition and the commutativity of flows conditions are ''independent'' of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.
==Intuitive meaning==
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In general, the quotient space is a bit difficult to work with when doing concrete calculations (not to mention nonlocal when working with [[diffeomorphism constraint]]s), so what is usually done instead is something similar. Note that the restricted submanifold is a [[Bundle (mathematics)|bundle]] (but not a [[fiber bundle]] in general) over the quotient manifold. So, instead of working with the quotient manifold, we can work with a [[Section (category theory)|section]] of the bundle instead. This is called [[gauge fixing]].
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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==Examples==
Consider the dynamics of a single point particle of mass {{mvar| m}} with no internal degrees of freedom moving in a [[pseudo-Riemannian]] spacetime manifold {{mvar|S}} with [[metric tensor|metric]] '''g'''. Assume also that the parameter {{mvar|τ}} describing the trajectory of the particle is arbitrary (i.e. we insist upon [[
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 {{mvar|H}} is, surprisingly enough, {{mvar|H}} = 0. In light of the observation that the Hamiltonian is only defined up to the equivalence class of smooth functions agreeing on the constrained subspace, we can use a new Hamiltonian {{mvar|H}} '= {{mvar|f}} instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See [[Hamiltonian constraint]] for more details.
Consider now the case of a [[Yang–Mills theory]] for a real [[simple Lie algebra]] {{mvar|L}} (with a [[negative definite]] [[Killing form]] {{mvar|η}}) [[minimally coupled]] to a real scalar field {{mvar|σ}}, which transforms as an [[orthogonal representation]] {{mvar|ρ}} with the underlying vector space {{mvar|V}} under {{mvar|L}} in (
:{{math|''ρ(l)[σ]''}}
as
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{{mvar|ρ}} ' is the dualized intertwiner
:<math>\rho':\bar{V}\otimes V\rightarrow L</math>
(
:<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
In a constrained Hamiltonian system, a dynamical quantity is '''second
See
===An example: a particle confined to a sphere===
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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>}}.
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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}
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Before analyzing the Hamiltonian, consider the three constraints,
:<math>
\
</math>
Note the nontrivial [[Poisson bracket]] structure of the constraints. In particular,
:<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, {{math| ''φ''<sub>2</sub>}} and
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
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
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
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}} as a primary constraint and proceed through the formalism: The result would the elimination of the extraneous {{mvar|λ}} dynamical quantity. However, the example is more edifying in its current form.
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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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==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 =
* {{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 =
[[Category:Classical mechanics]]
[[Category:Theoretical physics]]
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