This chapter introduces the classical equations of motion for a system of molecules, and describes their solution by stable, accurate, time-stepping algorithms. Simple atomic systems, rigid molecules, and flexible molecules with and without constraints, are treated, with examples of program code. Quaternions are introduced as useful parameters for solving the rigid-body equations of motion of molecules. A simple example of a multiple timestep algorithm is given, and there is a brief summary of event-driven (hard-particle) dynamics. Examples of constant-temperature molecular dynamics using stochastic and deterministic methods are presented, and the corresponding constant-pressure molecular dynamics methods for fixed and variable box-shape are described. The molecular dynamics method is extended to the treatment of polarizable systems, and dynamical simulation of the grand canonical ensemble is mentioned.
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