Answer by Qmechanic for Hamiltonian from a Lagrangian with constraints?

Comments to the question (v2):

To go from the Lagrangian to the Hamiltonian formalism, one should perform a (possible singular) Legendre transformation. Traditionally this is done via the Dirac-Bergmann recipe/cookbook, see e.g. Refs. 1-2. Note in particular, that the constraint $f$ may generate a secondary constraint

$$g ~:=~ \{f,H^{\prime}\}_{PB} +\frac{\partial f}{\partial t}~\approx~\frac{d f}{d t}~\approx~0.$$

[Here the $\approx$ symbol means equality modulo eqs. of motion or constraints.]

References:

1. P.A.M. Dirac, Lectures on QM, (1964).

2. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.