# Symplectic Runge-Kutta methods

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.

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?

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.
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.