  In this paper a geometric field theory of dislocation dynamics and finite
plasticity in single crystals is formulated. Starting from the multiplicative
decomposition of the deformation gradient into elastic and plastic parts, we
use Cartan's moving frames to describe the distorted lattice structure via
differential $1$-forms. In this theory the primary fields are the dislocation
fields, defined as a collection of differential $2$-forms. The defect content
of the lattice structure is then determined by the superposition of the
dislocation fields. All these differential forms constitute the internal
variables of the system. The evolution equations for the internal variables are
derived starting from the kinematics of the dislocation 2-forms, which is
expressed using the notions of flow and of Lie derivative. This is then coupled
with the rate of change of the lattice structure through Orowan's equation. The
governing equations are derived using a two-potential approach to a variational
principle of the Lagrange-d'Alembert type. As in the nonlinear setting the
lattice structure evolves in time, the dynamics of dislocations on slip systems
is formulated by enforcing some constraints in the variational principle. Using
the Lagrange multipliers associated with these constraints, one obtains the
forces that the lattice exerts on the dislocation fields in order to keep them
gliding on some given crystallographic planes. Moreover, the geometric
formulation allows one to investigate the integrability -- and hence the
existence -- of glide surfaces, and how the glide motion is affected by it.
Lastly, a linear theory for small dislocation densities is derived, allowing
one to identify the nonlinear effects that do not appear in the linearized
setting.
