First-class constraint: Difference between revisions

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InA a'''first constrainedclass Hamiltonianconstraint''' system,is a dynamical quantity is calledin a '''firstconstrained class[[Hamiltonian constraint'''mechanics|Hamiltonian]] ifsystem itswhose [[Poisson bracket with all the other constraints]] vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints) with all the other constraints. A '''second class constraint''' is one that is not first class.

First and second class constraints were introduced by {{harvs|txt|last=Dirac|authorlink=Paul Dirac|year1=1950|loc=p.136|year2=1964|loc2=p.17}} as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.<ref>{{Citation | last1=Dirac | first1=P. A. M. | author1-link=Paul Dirac | title=Generalized Hamiltonian dynamics | doi=10.4153/CJM-1950-012-1 |mr=0043724 | year=1950 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=2 | pages=129–148}}</ref>

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==Poisson brackets==

In [[Hamiltonian mechanics]], considerConsider a [[symplectic manifold]] ''M'' with a [[smooth function|smooth]] Hamiltonian over it (for field theories, ''M'' would be infinite-dimensional).

Suppose we have some constraints