Abstract: Hamel's field variational integrators are numerical schemes for classical field theory. It reduces computational cost caused by geometrical nonlinearity and exhibits a long-term energy stability and momentum-preserving property numerically. In the framework of one-dimensional field theory, taking geometrically exact beam as an example, this paper investigates theoretically discrete momentum conservation law of Hamel's field variational integrator. The major studies of this paper include the following aspects: The dynamical model of geometrically exact beam is established by using moving frame methods, dynamical equations of geometrically exact beam are obtained by variational principle, a momentum conservation law is then obtained through its dynamical equations and Noether theorem; For discrete model of geometrically exact beam, a discrete momentum conservation law is given by utilizing Hamel's field variational integrator of geometrically exact beam and discrete Noether theorem, and then the first order approximation of discrete momentum is proposed. Hamel's field variational integrators use system's symmetry to simplify the geometrical nonlinearity. It locates discrete convective velocities, discrete convective strain and configurations at different nodes on the spatial-temporal grid, thus leading to a series term in the expression of discrete momentum. This paper discusses the relation between the expression of continuous and the corresponding discrete one. Analytical and numerical examples are proposed to verify the conclusion. The proposed proof above is also applicable to the case in classical field theory and motivates further investigation of structure-preserving properties of Hamel's field variational integrator.