In particular, suppose a complex system is parameterized by a (large) set of coordinates, and the equations of motion are such as to minimize an expression in the coordinates and their first time derivatives as is typical of Lagrangians in physics; I'll simply refer to it as the Lagrangian going forward, though in some contexts it might be a thermodynamic potential or a utility function or the like. A good division amounts to separating the coordinates into three subsets, say A, B, and C, where A and C at least are nonempty and the (disjoint) union of the three is the full set of coordinates of the system. Given values of the coordinates (and their derivatives) in A and B, we can calculate the optimal value of the Lagrangian by optimizing over values that C can take, and similarly given B and C I can optimize over A, and I get effective Lagrangians for (A and B) and (B and C) respectively. Where this works best, though, is where the optimizing coordinates in C (in the first case) or A (in the second) depend only on the coordinates in B; conditional on B, A and C are independent of each other. This works even better if B is fairly small compared to both A and C, and might in that case even be quite useful if the conditional independence is only approximate.

In general there will be many ways to write the Lagrangian as L

_{A}+L

_{B}+L

_{C}+L

_{AB}+L

_{BC}+L

_{AC}+L

_{ABC}, with each term depending only on coordinates in that set, but it will be particularly useful to write a Lagrangian this way if the last two terms are small or zero. If we are "optimizing out" the C coordinates, the effective Lagrangian for A and B is L

_{A}+L

_{B}+L

_{AB}plus what we get from optimizing L

_{C}+L

_{BC}+L

_{AC}+L

_{ABC}; this will depend only on B if the terms with both A and C are absent. Thus on some level a good decomposition of a system is one in which the Lagrangian can be written as L

_{AB}+L

_{B}+L

_{BC}, where I've absorbed previous L

_{A}and L

_{C}terms into the first and last terms; for given evolutions of B variables, the A variables will optimize L

_{AB}and the C variables will optimize L

_{BC}and these two optimizations can be done separately.

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