For broader coverage of this topic, see Dirac bracket.
In physics, a first class constraint is a dynamical quantity in a constrained 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 brackets generated.[1]
First and second class constraints were introduced by Dirac (1950, p.136, 1964, p.17) as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.[2][3]
The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.
^Ingemar Bengtsson, Stockholm University. "Constrained Hamiltonian Systems" (PDF). Stockholm University. Retrieved 29 May 2018. We start from a Lagrangian L ( q, ̇ q ), derive the canonical momenta, postulate the naive Poisso n 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 [...]
^Dirac, Paul A. M. (1950), "Generalized Hamiltonian dynamics", Canadian Journal of Mathematics, 2: 129–148, doi:10.4153/CJM-1950-012-1, ISSN 0008-414X, MR 0043724, S2CID 119748805
^Dirac, Paul A. M. (1964), Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, vol. 2, Belfer Graduate School of Science, New York, ISBN 9780486417134, MR 2220894. Unabridged reprint of original, Dover Publications, New York, NY, 2001.
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