In a constrained Hamiltonian system, a dynamical quantity is called a first class constraint if its Poisson bracket with all the other constraints vanishes on the constraint surface (the surface implicitly defined by the simultaneous vanishing of all the constraints). A second class constraint is one that is not first class.
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.
The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints. 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.
Read more about First Class Constraint: Poisson Brackets, Geometric Theory, Intuitive Meaning, Constrained Hamiltonian Dynamics From A Lagrangian Gauge Theory, Examples, Second Class Constraints
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