Gaussian, or isokinetic, dynamics was introduced by
Hoover and Evans in 1982. This is the idea:
Let
(7.1)
be a (non-holonomic) constraint to the Newtonian motion of the
-particle system. This is only one constraint equation for
variables, thus we have some freedom of choice.
A dynamical constraint may always be put in terms of
constraint forces. A particularly economical and ``impartial'' prescription for the definition of these forces was given by
C. F. Gauss in 1829:
From
for all times we have
(7.2)
Since
(with the Newtonian forces and the yet unknown constraint
forces ) we have
(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.