A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...

with all the other constraints vanishes on the constraint surface in

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...

(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 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.

Poisson brackets

Consider a

Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalen ...

''M'' with a smooth Hamiltonian over it (for field theories, ''M'' would be infinite-dimensional). Suppose we have some constraints :

f_i(x)=0,

for ''n'' smooth functions :

\_^n

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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...

and also that the