Trond Ingebrigtsen Glass and Time Roskilde University , Denmark 1 Constrained Dynamics : TSCD algorithm

where λ is the Lagrange multiplier and ψ = 0 the holonomic constraint. The first practical algorithm for constrained dynamics was called SHAKE [3]. In this algorithm the Lagrange multiplier was calculated from the viewpoint, that no exact discrete solution could be obtained. Thus the constraints were satisfied by inserting the discrete propagator into the holonomic constraint, ψ(t+ h) = 0. This algorithm is entirely stable, but the calculation of λ resulted in solving quadratic equations. Edberg et al. [4] introduced a new paradigm by calculating the Lagrange multiplier from the time derivative ψ̈(t) = 0, and applying a linearization to vij(t). This resulted in solving linear equations in λ . This linearization unfortunately violates the time reversal invariance, and the papers by Toxvaerd et al. [1] and Ingebrigtsen et al. [2] investigate this fundamental problem in different ensembles. Their work also discovered an entirely new connection between the method of Edberg et al. and the SHAKE algorithm.