Exactly conservative integrators

Typically when numerically integrating a system of differential equations, nonlinear constants of motion possessed by the system are not conserved by the numerical method. We present a nontraditional algorithm that exactly conserves a systems' constants of motion to all orders. In contrast to traditional numerical methods, the time advance algorithm described here is not a polynomial function of the time step. We show that this is a generic requirement of an exactly conservative integrator. We apply our method to turbulent fluids simulations as well as to the motion of a point particle in an external electromagnetic field. We discuss further applications and extensions of this method