  The use of Lagrange multipliers in the context of quintessence/phantom scalar
fields allows to constrain the behavior of the scalar field, which provides a
powerful tool, not only for the reconstruction of cosmological solutions but
also for the study of some problems in cosmology and gravitational physics. In
the present paper, we focus on the reconstruction of cosmological solutions
capable of controlling the cosmological constant value by imposing a constraint
on the scalar field, providing a relaxation mechanism of the value of the
cosmological constant. The formalism is also extended to the study of phantom
scalar fields with a future singularity and their conformal transformation to
the Jordan frame, where a type of modified gravity, constrained by the Lagrange
multiplier, is obtained.
