  Classical energy-momentum methods study the existence and stability
properties of solutions of $t$-dependent Hamilton equations on symplectic
manifolds whose evolution is given by their Hamiltonian Lie symmetries. The
points of such solutions are called relative equilibrium points. This work
devises a new cosymplectic energy-momentum method providing a new and more
general framework to study $t$-dependent Hamilton equations. In fact,
cosymplectic geometry allows for using more types of distinguished Lie
symmetries (given by Hamiltonian, gradient, or evolution vector fields),
relative equilibrium points, and reduction methods, than symplectic techniques.
To make our work more self-contained and to fill some gaps in the literature, a
review of the cosymplectic formalism and the cosymplectic Marsden-Weinstein
reduction is included. Known and new types of relative equilibrium points are
characterised and studied. Our methods remove technical conditions used in
previous energy-momentum methods, like the ${\rm Ad}^*$-equivariance of
momentum maps. Eigenfunctions of $t$-dependent Schr\"odinger equations are
interpreted in terms of relative equilibrium points in cosymplectic manifolds.
A new cosymplectic-to-symplectic reduction is developed and a new associated
type of relative equilibrium points, the so-called gradient relative
equilibrium points, are introduced and applied to study the Lagrange points and
Hill radii of a restricted circular three-body system by means of a not
Hamiltonian Lie symmetry of the system.
