Gauss’s principle of least constraint

D.J. Evans, W.G. Hoover, B.H. Failor, B. Moran, and A.J.C. Ladd (1983) Non-equilibrium molecular dynamics via Gauss s principle of least constraint. Phys. 

The present treatment covers mainly general holonomic constraints, with only a brief mention of nonholonomic constraints in the context of the application of Gauss s principle of least constraint in MD simulations. A holonomic constraint, a = 0, is defined as an algebraic equation connecting the coordinates of the particles (and in general also the time, but not here). To completely understand the various constraint methods used in modem simulations, it is appropriate to start the discussion from first principles. 

In the EEM method. Gauss s principle of least constraint is invoked to derive the equations of motion of the system of particles with holonomic constraints. However, it is well known that when holonomic constraints are involved, the equations of motion can be derived from either D Alembert s principle, Hamilton s principle, or by means of a third approach. Application of Gauss s principle in this case offers no advantage over these other approaches. Gauss s principle is also exploited in the EEM method to enforce a nonholonomic temperature constraint in constant-temperature MD simulations. Again, the same equations of motion can be obtained by alternative means. ... 

Nonequilibriura Molecular Dynamics via Gauss s Principle of Least Constraint. 

To formulate a constrained model we use Gauss s principle of least constraint which says that the force needed to compensate for the presence of a constraint of the form... 

Since the equations of motion (9.37) constrain T, they are generally closer to experiment than the traditional Newtonian equations (i.e., with a = 0) which constrain the total energy. The particular functional form of the thermostat (9.37) and (9.40) is derived using Gauss s Principle of Least Constraint. Today, however, several thermostatting schemes have been proposed and the choice is one of convenience. (See Evans Morriss 1990 and Allen Tildesley 1990 for a detailed discussion of thermostats.)... 

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