I believe it used to be standard to use Lagrange multipliers to handle constraints on the state space in R^n as an alternative to using generalized coordinates. For example, when you handle the case of motion on the surface of a sphere you can either use R^3 with a Lagrange multiplier enforcing the constraint that you stay on the surface of the sphere, or you could use latitude/longitude as coordinates.

The interpretation of the momenta themselves as Lagrange multipliers is completely nonstandard, though. Try googling "hamiltonian mechanics lagrange multipliers"; you'll get either examples like the sphere thing or hits from optimal control tutorials.