Coupled second-order differential equations using runge kutta 45

Question

As a little summer project I have tried to make a ballistic calculator for when I play football, (following an example from a book), just to learn some numerical methods while doing so. My problem is that I cant seem to find much on the Runge-Kutta 5 method (method recommended) for systems of second order differential equations.

My equations (simplified a bit) is

$ \cfrac{d^2x}{dt^2} = K_x(y) \cdot \sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2} \cdot \dot{x} + C_x(y) $

$ \cfrac{d^2y}{dt^2} = K_y(y) \cdot \sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2} \cdot \dot{y} + C_y(y) $

$ \cfrac{d^2z}{dt^2} = K_z(y) \cdot \sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2} \cdot \dot{z} + C_z(y) $

With the initial conditions being

$\dot{x}(0)=v_{x0}$,
$\dot{y}(0)=v_{y0}$,
$\dot{z}(0)=v_{z0}$.

$x(0)=x_0$,
$y(0)=y_0$,
$z(0)=z_0$.

So far I have proceeded as I usually do when implementing the Runge-Kutta method up to this point (which is just the first step)

$ \cfrac{d^2x}{dt^2} = f(t, x, y, z, \dot{x}, \dot{y}, \dot{z}) $

$ \cfrac{d^2x}{dt^2} = f(t, x, y, z, \dot{x}, \dot{y}, \dot{z}) $

$ \cfrac{d^2x}{dt^2} = f(t, x, y, z, \dot{x}, \dot{y}, \dot{z}) $

How can I proceed from this point? Bonus if someone knows some good literature on the subject.

Answer 1

The reason you don't find something on integrators of higher order ODEs is that you can always convert them to first order ODEs with auxiliary variables.

A equation like $$ \frac{d^2x}{dt^2} = f(t, x, \dot{x}) $$ and $\dot{x}(0)=\dot{x}_0, x(0)=x_0$ can be converted to with an auxiliary variable $\xi$ $$ \frac{dx}{dt} = \xi \\ \frac{d\xi}{dt} = f(t, x, \xi) $$ with initial conditions $\xi(0)=\dot{x}_0$ and $x(0)=x_0$. Now, each equation is first order but you have two equations.

If you write this as a vector this leads to $$ \frac{d}{dt}\left(\begin{array}{c}x\\\xi\end{array}\right) = \left(\begin{array}{c}\xi\\ f(t,x,\xi)\end{array}\right)\\ $$ and you can pass this to your first order ODE integrator of your choice.