  We consider the Hamiltonian constraint formulation of classical field
theories, which treats spacetime and the space of fields symmetrically, and
utilizes the concept of momentum multivector. The gauge field is introduced to
compensate for non-invariance of the Hamiltonian under local transformations.
It is a position-dependent linear mapping, which couples to the Hamiltonian by
acting on the momentum multivector. We investigate symmetries of the ensuing
gauged Hamiltonian, and propose a generic form of the gauge field strength. In
examples we show how a generic gauge field can be specialized in order to
realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is
not discussed in this article. Throughout, we employ the mathematical language
of geometric algebra and calculus.
