SUBJECTS dynamic philosophy 164 dynamics

An alternative philosophy is to construct a classical trajectory which is not a solution of an initial value problem. Such constrained trajectories would no longer obey Hamilton s equations, but would extremize the classical action subject to certain dynamical constraints (50-52). 

For a concrete example, consider a crystalline solid subjected to external loading in the form of an indenter like the one shown in Figure 1. The quasicontinuum philosophy is to discretize the system in such a way that there is full atomic resolution where the action is (such as beneath the indenter) and a select, representative subset of atoms that serve as nodes of the finite-element regions where all-atom resolution is surrendered. For the particular case of two-dimensional dynamical nanoindentation considered here, the calculation involves a total of 5,000 nodes as opposed to the lO" atoms that would be needed in a full atomistic calculation. This point is driven home in an even more compelling fashion in the case of a fully three-dimensional calculation for which the full atomistic calculation would have implicated in excess of 10 atoms (Knap and Ortiz, 2003). 

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