## 7.1.1 Gauss dynamics

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.

F. J. Vesely / University of Vienna