Is there any explicit symplectic Runge-Kutta method?

Question

As far as I know, all the symplectic Runge-Kutta methods are implicit which need to solve non-linear equations during the calculation. Is there any explicit method? If not, why?

Answer 1

There are explicit, symplectic methods for certain types of Hamiltonian problems. For example, the symplectic Euler method

\begin{align} p_{n+1} &= p_n - h H_q(p_{n+1}, q_n) \\ q_{n+1} &= q_{n} + h H_p(p_{n+1}, q_n) \end{align}

is symplectic, see e.g. Theorem 3.3 on p. 189 in the book by Hairer, Wanner and Lubich (see full reference below). For simple functions $H$ like $H(p, q) = \frac{1}{2} \left( p^2 + q^2 \right)$, this becomes

\begin{align} p_{n+1} &= p_n - h q_n \\ q_{n+1} &= q_n + h p_{n+1} \end{align}

which is explicit. More generally, this method is explicit for separable Hamiltonians (see comments after the Theorem mentioned above).

In VIII.6 on p. 325, Hairer et al state that

Symplectic methods for general Hamiltonian equations are implicit, and so are symmetric methods for general reversible systems.

Therefore, there are no sympletic methods that are explicit for general Hamiltonian functions. I couldn't find the specific Theorem that states this, though.

Hairer, Ernst; Lubich, Christian; Wanner, Gerhard, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31. Berlin: Springer (ISBN 3-540-30663-3/hbk). xvii, 644 p. (2006). ZBL1094.65125.