PHY.5:4.4 - Construct the approximation and its first omitted contribution

Use MMP.9 for the mathematical elimination or retained-state evolution. Keep the physical assumptions that permit the construction alongside its result.

For a rapidly relaxing state, examine its initial departure and response to changing inputs. Replacing it by a steady response is useful when the remaining departure has sufficiently little effect on the requested consequence. The motor calculation below derives that departure instead of assigning it zero at the initial instant.

For a spatial or energy expansion, choose the ratio in which the description is expanded and compare the first omitted contribution with the retained ones. The ratio and its powers come from the physical relations. In the chain example, expansion in the small ratio of lattice spacing a to wavelength lambda, or equivalently in q*a=2*pi*a/lambda, yields a continuum wave description and its leading correction.

When unresolved correlations persist over the interval of interest, retain their delayed influence or add variables that reproduce it. Replacing a delayed response by an instantaneous one needs a comparison of that response time with the retained motion and driving. A mathematically equivalent auxiliary state is another way to calculate the memory; its interpretation as a material component requires a separate physical account.

Maintain the relevant exchanges and constraints through the approximation. Recover the condition for a balance or an allowed motion when changing the variables or boundaries. C.29.BB supplies the common balance construction.