First Class Constraint - Poisson Brackets

Poisson Brackets

In Hamiltonian mechanics, 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

{ f_i, f_j }

and

{ f_i, H }

all vanish on the constrained subspace. This means we can write

for some smooth functions

c_ijk

(there is a theorem showing this) and

for some smooth functions

v_ij.

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).