Numerically Integrating the SLLOD Equations

Let us consider the time evolution of phase space variable F, that is, 

Once again, F is an abstract vector in the full phase space (i.e., including extended system variables). The majority of this section is devoted to how one mathematically represents the Formally, this is quite simple. Suppose that we want to evolve a phase variable B(F). We can write differential equations for B  

Here again, we have introduced the Liouville operator for the phase variables and have also noted that the phase variable does not have any explicit time dependence. We can solve Eqs. [139] to obtain 

we have assumed that the Liouville operator has no explicit time dependence. To proceed further, we need to specify the equations of motion so that we can explicitly write out L. The simplest nontrivial dynamics for an isolated system are Hamilton s equations of motion, which read 

q is the position, p is the conjugate momentum, m is the mass, and F is the force. Thus, the phase space vector T has two components  

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