First-class constraint: Difference between revisions

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A '''first class constraint''' is a dynamical quantity in a constrained [[Hamiltonian mechanics|Hamiltonian]] system whose [[Poisson bracket]] vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints) with all the other constraints. To calculate the first class constraint, we assume 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=Stockholm University|title=Constrained Hamiltonian Systems|url=http://www.fysik.su.se/~ingemar/Nr13.pdf|publisher=Stockholm University|accessdate=18 September 2015|page=7|language=English|format=PDF|quote=We start from a Lagrangian L ( q, ̇ q ), derive the canonical momenta, postulate the naive Poisso n brackets, and compute the Hamiltonian. For simplicity, we assume 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 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>