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Constraint Solvers Δt 2 f n + g n rn +1 = rn + Δt vn + + O Δt 3 2 m Δt f n + hn vn +1 = vn + 1 + + O(Δt 3 ) 2 2 m ( ) Taylor expansions: SHAKE iu ! Gij i io !u !d ij d ij j !o o d ij j !o ! ! Gij = − G ji ≈ g ij d ij ! 2 !u2 µij (d ij − d ij ) !o !u g ij = 2 2Δt d ij ⋅ d ij 1 µij = 1 1 + mi m j RATTLE RATTLE_R (SHAKE) ! G ji !o2 !u 2 µij (d ij − d ij ) g ij = 2 ! o ! u Δt d ij ⋅ d ij ju ! vio i j ! d ij RATTLE_V ! vio !o ! ! H ij = − H ji = hij ⋅ d ij ! ! !o µij (vi − v j ) ⋅ d ij hij = Δt d ij2
