First class constraint
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.
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, 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.
Consider a symplectic manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite-dimensional).
Suppose we have some constraints
for n smooth functions
These will only be defined chartwise in general. Suppose that everywhere on the constrained set, the n derivatives of the n functions are all linearly independent and also that the Poisson brackets
and
all vanish on the constrained subspace.
This means we can write
for some smooth functions −−there is a theorem showing this; and
...
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