Boundary conditions brute force

There are the usual boundary conditions depending on the experiment performed on this system. One possible way to handle all this is simply to write out the whole system as a large linear system, expand that to include the boundary conditions, and solve. This, brute force approach (see below), has in fact been used [138] and can even be reasonably efficient if the number of equations is kept low, by use, for example, of imequal intervals, described in Chap. 7. If the equations in such a system are arranged in the order as above (6.55), it will be found that it is tightly banded, except for the first two rows for the boundary conditions, which may have a number of entries up to the number n used for the current approximation. 

Molecular dynamics is used to simulate the motions in many-body systems. By MD we mean the brute force solution of the classical equations of motion subject to pertinent boundary conditions. In a typical MD simulation one first starts with an initial state of the system. This is a list of the coordinates required to define the positions of all of the particles in the system and the momenta conjugate to the.se coordinates. Thus for an N particle system there will be 3N coordinates and 3N conjugate momenta and the mechanical state of the system will thus be specified by a 6N-dimensional vector r = x, X2,X3N, p, P2> PiN)- After sampling the initial state one must numerically solve Hamilton s equations of motion ... 

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