Lagrange multipliers are a common technique in classical mechanics for imposing constraints on dynamical variables. These are often trivial to include in a Lagrangian, but much harder to deal with in a Hamiltonian formulation. In fact, this is one of the many issues which make gauge theories complicated to handle.
What I would like to describe here is a very nice way to relate Lagrange multipliers in the Lagrangian formalism with delta functions in the path integral.
Suppose we want to think about a quantum system which contains some fields \(\phi_a\) and action \(S\). Then we know the generating functional with sources \(J^a\) is given by
$$Z[J]=\int[\mathcal{D}\phi]e^{iS+i\int\text{d}^dx J^a\phi_a}$$
in Lorentz signature and with the summation convention enforced.
If we wanted to add some constraint to our system, say \(\phi_a\phi_a=r\) where \(r\) is some constant, we know to add a Lagrange multiplier we would add a new field \(\lambda\) and a term \( \lambda (\phi_a \phi_a-r) \) to the Lagrangian. The generating functional for this constrained system would then be
$$Z[J]=\int[\mathcal{D}\phi\mathcal{D}\lambda]\text{Exp}\left[iS+i\int\text{d}^dx J\phi+i\int\text{d}^dx \lambda(\phi_a\phi_a-r)\right],$$
where we have written the Lagrange multiplier term separately from the rest of the action.
But now remember that the Fourier transform of the Dirac delta function may be written
$$\delta^d(x)=\int\frac{\text{d}^dk}{(2\pi)^d}e^{ix\cdot k}.$$
The functional equivalent of this identity,
$$\delta[f]=\int[\mathcal{D}\lambda]\text{Exp}\{i\int\text{d}^dx \lambda(x)f(x)\},$$
in which the numerical prefactor has been ignored, looks rather like the Lagrange multiplier term we added to impose the constraint. In our case, however, we have \(f=\phi_a\phi_a-r\).
This all means that we can do the \(\lambda\) integral to write the Lagrange multiplier as a delta functional,
$$Z[J]=\int[\mathcal{D}\phi]\delta[\phi_a\phi_a-r]e^{iS+i\int\text{d}^dxJ^a\phi_a},$$
which now gives the mechanical process of adding a Lagrange multiplier to our Lagrangian the obvious interpretation of forcing our path integral to consider only those field configurations which obey the constraint.
This same sort of reasoning is in essence what underpins the Faddeev-Popov method for gauge-fixing Yang-Mills gauge theories.
