Runge--Kutta methods determined from extended phase space methods for Hamiltonian systems

We study two existing extended phase space integrators for Hamiltonian systems, the {\em midpoint projection method} and the {\em symmetric projection method}, showing that the first is a pseudosymplectic and pseudosymmetric Runge–Kutta method and the second is a monoimplicit symplectic Runge–Kutta method.