Lattice gauge theory, both in its Euclidean and Hamiltonian version, has been described in the literature exhaustingly[6]. Here we deal only with the classical Hamiltonian SU(2) theory specified by a Hamilton function, which describes a coupled system of quaternionic pendulums
where the summation runs over all oriented links of the three dimensional cartesian lattice, and the usual magnetic energy form
containing a summation of traces of oriented products 
of four group
elements U on links circumventing a plaquette --- an elementary square
of the lattice --- is rewritten as a sum over all links of the scalar
product of link variables U with their complement V defined by the
sum of all triple products of oriented link group elements closing a
plaquette with the chosen link. Since this way all plaquettes are counted
four times in the sum --- because each plaquette is surrounded by four
links --- a division by 4 completes the definition of V. This is
no more a group element, because its length is no more unity. It can, however,
be written as its quaternionic length 
multiplied by a group
element. This way we describe a system of coupled pendulums each of which
feels a "gravity" caused by some neighbouring ones.
The equations of motion following from the above Hamiltonian has to be
approximated by a Type III algorithm described in
eq.( 39 ) with
an additional force 
,
with the Lagrange multipliers
This update equations conserve then the norm, the orthogonality and the Noether charge calculated via the respective half integer time definitions. The energy, however, is not conserved anymore locally.
