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A first class constraint is a dynamical quantity in a constrained 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 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. 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 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 : for some smooth functions :. This can be done globally, using a partition of unity. Then, we say we have an irreducible first-class constraint (''irreducible'' here is in a different sense from that used in representation theory). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「first class constraint」の詳細全文を読む スポンサード リンク
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