Constraints and numerical integration

In an autonomous Hamiltonian system, one constraint always exists, namely the energy integral or a constant magnitude of the 4-velocity in relativistic dynamics. The constraint should bring better numerical stability if it can be kept at every step of the numerical integration. In Newtonian mechanics, the order of the equations of motion can not be reduced by use of the constraint in most cases, because its kinetic energy is usually a quadratic form of the elliptic type and one would meet difficulty when trying the order reducing. However, the metric in general relativity is hyperbolic. In particular, when the spacetime bears some symmetries there exists a global transformation so that at least one element in its main diagonal vanishes. As a result, the constraint can be solved for a certain velocity or a momentum without any difficulty, and so reducing the order. Similarly, this technique can also be applied to the evolution of the Mixmaster universe. It is shown that this technique can raise the precision and improve the numerical stability dramatically even when a classical integrator is used, although it might not keep the symplectic structure of the system.