  We study scalar-tensor theory, k-essence and modified gravity with Lagrange
multiplier constraint which role is to reduce the number of degrees of freedom.
Dark Energy cosmology of different types ($\Lambda$CDM, unified inflation with
DE, smooth non-phantom/phantom transition epoch) is reconstructed in such
models. It is shown that mathematical equivalence between scalar theory and
$F(R)$ gravity is broken due to presence of constraint. The cosmological
dynamics of $F(R)$ gravity is modified by the second $F_2(R)$ function dictated
by the constraint. Dark Energy cosmology is defined by this function while
standard $F_1(R)$ function is relevant for local tests (modification of newton
regime). A general discussion on the role of Lagrange multipliers to make
higher-derivative gravity canonical is developed.
