Higher Order Symplectic Methods SHAKE and RATTLE

5 Higher Order Symplectic Methods SHAKE and RATTLE 

The projected symplectic constrained method (4.20)-(4.24) is only first order accurate. We forego providing a detailed proof of this fact, but note that it could be demonstrated using standard methods [164]. Note that (4.20)-(4.24) reduces to the symplectic Euler method in the absence of constraints, and the projection of the momenta would not alter this fact. There are several constraint-preserving, second-order alternatives which generalize the Stormer-Verlet scheme. One of these is the SHAKE method [322]. The original derivation of the SHAKE method began from the position-only, two-step form of the Stormer rule for q = F(q) 

The symmetry suggests that this will be second order. In order to understand the symplectic property associated to this method, we need to define the updates for both positions and momenta. A natural choice is to consider the phase space formulation of the Stormer-Verlet method for q = M p,p = F q), then replace F by F — 

It is possible to demonstrate (see exercises) that this method reduces to (4.25)-(4.26) by eliminating the momenta. However (4.27)-(4.29) does not give a map of the cotangent bundle, since g qn+i)Pn+i = 0 will not be preserved, even if both the constraint and hidden constraint hold at step n. 

A method that automatically preserves both the constraint and hidden constraint is the RATTLE method [12] which consists of the following steps  

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