Geometric integrator

Remarks

Example: take the harmonic oscillator $q' = p$, $p' = -q$. The symplectic Euler method $p_{n+1} = p_n - h q_n$, $q_{n+1} = q_n + h p_{n+1}$ is the linear map $$\begin{pmatrix} q_{n+1} \\ p_{n+1} \end{pmatrix} = \begin{pmatrix} 1 - h^2 & h \\ -h & 1 \end{pmatrix} \begin{pmatrix} q_n \\ p_n \end{pmatrix}.$$ Its determinant is $1$, and in dimension $2$ this means that the map is symplectic. So the method is a geometric integrator for property 1. Non-example: the explicit Euler method $q_{n+1} = q_n + h p_n$, $p_{n+1} = p_n - h q_n$ has matrix determinant $1 + h^2 \ne 1$, so it is not symplectic for any $h > 0$. A method may be geometric for one property and not another. The implicit midpoint rule is symplectic and conserves all quadratic first integrals. But no Runge–Kutta method conserves all polynomial first integrals of degree $3$ or more.