Let

(7.1) |

A dynamical constraint may always be put in terms of

- From
for all times we have

(7.2)

(7.3) - Among all possible sets of constraint forces let us choose the one with
the smallest norm. Requiring
we find, upon variation, that
. Since
for all physically allowed variations we have
we conclude that
. Inserting this in
we find

(7.4) - The equations of motion according to Gauss and Hoover are thus

(7.5) (7.6)

These e.o.m. lead to a deterministic trajectory with .

Note that and are needed at the same time step; thus the usual Verlet algorithm is not applicable here. Instead, a Predictor-Corrector algorithm or Runge-Kutta should be used.

F. J. Vesely / University of Vienna