  We first consider the Einstein-aether theory with a gravitational coupling
and a Lagrange multiplier field, and then consider the non-minimally coupled
quintessence field theory with Lagrange multiplier field. We study the
influence of the Lagrange multiplier field on these models. We show that the
energy density evolution of the Einstein-aether field and the quintessence
field are significantly modified. The energy density of the Einstein-aether is
nearly a constant during the entire history of the Universe. The energy density
of the quintessence field can also be kept nearly constant in the matter
dominated Universe, or even exhibit a phantom-like behavior for some models.
This suggests a possible dynamical origin of the cosmological constant or dark
energy. Further more, for the canonical quintessence in the absence of
gravitational coupling, we find that the quintessence scalar field can play the
role of cold dark matter with the introduction of a Lagrange multiplier field.
We conclude that the Lagrange multiplier field could play a very interesting
and important role in the construction of cosmological models.
