We discuss how dynamical fermion computations may be made yet cheaper byusing symplectic integrators that conserve energy much more accurately withoutdecreasing the integration step size. We first explain why symplecticintegrators exactly conserve a ``shadow'' Hamiltonian close to the desired one,and how this Hamiltonian may be computed in terms of Poisson brackets. We thendiscuss how classical mechanics may be implemented on Lie groups and derive theform of the Poisson brackets and force terms for some interesting integratorssuch as those making use of second derivatives of the action (Hessian or forcegradient integrators). We hope that these will be seen to greatly improveenergy conservation for only a small additional cost and that their use willsignificantly reduce the cost of dynamical fermion computations.
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