Symplectic Runge-Kutta methods

Question

We can solve some differential equation numerically using Runge-Kutta methods. i.e. for some function $f(x,y) = \frac{dy}{dx}$, then

$$ y_{n+1} = y_n + \sum_{i=1}^{s} c_i k_i $$

where e.g.

$$ k_2 = f(x_n+a_2 h, y_n + b_{21}k_1)$$

and $a_i$, $b_{ij}$ are some coefficeints e.g. Cash-Karp.

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Now, I understand that there also exist symplectic Runge-Kutta methods. How would this method work? What would be the equivalent $y_{n+1}$ style equations for a symplectic Runge-Kutta? How are the coefficients chosen?

Answer 1

The following is sufficient and necessary for a Runge-Kutta method to be symplectic: $$ b_i a_{i,j} + b_j a_{j,i} - b_ib_j = 0 \, \quad 1 \leq i,j \leq s $$ where $s$ is the number of stages in your method. No explicit RK methods achieve this.

This condition is in addition to other consistency conditions too.

Also, don't forget the $dt$ in your method, and the convention is to use $b_i$ in the linear combination of stage derivatives $k_i$, $c_i$ multiplied by $h=dt$, and $a_{i,j}$ for the states.

source: http://www.sanzserna.org/pdf/56_physicaD.pdf

EDIT: As pointed out in the comments, this applies to symplectic Runge-Kutta methods, but generally methods may be symplectic but not Runge-Kutta.