  Classical field theory is considered as a theory of unparametrized surfaces
embedded in a configuration space, which accommodates, in a symmetric way,
spacetime positions and field values. Dynamics is defined via the (Hamiltonian)
constraint between multivector-valued generalized momenta, and points in the
configuration space. Starting from a variational principle, we derive the local
equations of motion, that is, differential equations that determine classical
surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field
theory then follows readily. In addition, we discuss the relation between
symmetries and conservation laws, and derive a Hamiltonian version of the
Noether theorem, where the Noether currents are identified as the classical
momentum contracted with the symmetry-generating vector fields. The general
formalism is illustrated by two examples: the scalar field theory, and the
string theory.
  Throughout the article, we employ the mathematical formalism of geometric
algebra and calculus, which allows us to perform completely coordinate-free
manipulations.
