Gauss’ principle

The external field does work on the system. This work is converted to heat which must be removed if one wants to reach a steady state. This can be done by applying a thermostat. Mathematically this is achieved by using Gauss principle of least constraint [13]. This is a powerful but not very well-known principle of mechanics that can be used to handle various kinds of constraints in a way similar to the application of the Lagrange equation. Gauss principle is based on a quantity called the square of the curvature, C,... 

It is regarded as a function of the linear and angular accelerations, (a, p ) are treated like constant parameters. The linear acceleration is denoted by a,-, and here it is assumed to be the rate of change of the peculiar momentum, a, = p,- /m. According to Gauss principle the equations of motion are obtained when C is minimal. It is immediately obvious that when the external field is equal to zero, C is minimal when each term in the sum is equal to zero so that Newton s and Euler s equations are recovered. 

We finally note that is possible to use Gauss principle to obtain equations of motion when the system is subject to holonomic constraints such as bond length or bond angle constraints. In this case one obtains the same equations of motion as one would obtain by applying the Lagrange equation. 

It will stay constant if is constrained to be constant. Applying Gauss principle gives... 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

Here, is the desired temperature and T is the current temperature of the system. In the Gauss thermostat [83], a friction is added to the interatomic forces. The friction term is derived from Gauss Principle of Lease Constraint, which maintains that the sum of the squares of any constraining forces on a system should be as small as possible. The friction force on each atom i is written... 

Gauss principle can be applied to yield the isokinetic ensemble of the combined system with the following equations of motion ... 

Constant pressure MD can be implemented either by extended system methods that couple the dynamical system to an external variable V which is the volume of the simulation box (Andersen, 1980) or by constraint methods that use a Lagrange multiplier determined from Gauss principle (Evans and Morriss,... 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

A set of linear equations can be solved by a variety of procedures. In principle the method of determinants is applicable to any number of equations but for large systems other methods require much less numerical effort. The method of Gauss illustrated here eliminates one variable at a time, ends up with a single variable and finds all the roots by a reverse procedure. 

We are now in a position to devise a first, very crude program that should, starting from a set of initial guesses, move towards the best fit. Below, a flow diagram is given that represents the basic principle of the Newton-Gauss algorithm ... 

Although it is, in principle, possible to evaluate (6.2.11) analytically, it is numerically more advantageous to evaluate the integral in (6.2.11) with the help of a suitable Gauss-Laguerre integration formula (see, e.g., Stroud and Secrest (1966)). 

The Gauss map of an IPMS (which is a function of the surface orientation only through the normal vectors) must be periodic, since a translationally periodic surface is necessarily orientationally periodic. (The converse, however, is not true.) Consequently, the Gauss map of IPMS must lead to periodic tilings of the sphere. This principle has been used to construct all the simpler IPMS, and has recently been generalised to allow explicit parametrisation of more complex "irregular" IPMS [13-24]. Some of these examples are illustrated in the Appendix to this chapter. 

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