Tobias and Brooks in Context

beginning with the starting point of the matrix method, Eq. [56], taking to be the subset of particles involved in a, and using Eq. [40] to Tay-lor-expand a ( r(to + 8t,[ y)))) about the preceding time step particle positions r(ro)l (rather than about the current time step unconstrained particle positions [r (rQ -I- 8t)]), Eq. [56] becomes 

r ( r(fo) ) = 0 because the constraints are satisfied (within the desired tolerance) at every time step and in particular at the preceding time step t. Replacing [V (r fe]( o) by the Wilson vectors and neglecting all nonlin- 

Equation [80] is equivalent to Eq. [IS] of the TB method. In the Wilson vector notation, the total displacement Eq. [55] can be written as follows  

Taylor-expanding CT ([r (tg + 8t) ) about the preceding time step particle positions r(t(j), we have 

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