Pinball scattering is a purely Newtonian equations of motion exercise without feedback or random numbers but showing chaotic behaviour regardless.

Consider the potential

$(,)=22(2+2)$

$($all terms dimensionless$)$ with four maxima centred at $=\backslash$pm $1$ and $=\backslash$pm $1,$ respectively. It is defined as a repulsive potential for any test particle $($the pinball$)$ moving through that potential. The equations of motion to solve hence are

$2()2=2\mathrm{.}02(12)(2+2)$

and

$2()2=2\mathrm{.}02(12)(2+2).$

Task: Solve these two coupled ODE's using solve$\_$ivp in two stages:

$($a$)$ write a function trajectory$($impactpar$,$ speed$)$ which takes the impact parameter, $(0),$ as input as well as the speed $$ for initial conditions $($impact parameter $=(0),=(0),(0)$ and $=(0)$ in that order$).$ Calculate the time array to solve for from np$.$linspace$(0,$maxtime,$300)$ where maxtime works well as $10/$ speed. The function shall return the trajectory, i$.$e$.$ the arrays of solved x$($t$)$ and y$($t$)$ separately. Plot a trajectory of your choice, see below, with appropriate axes labels.

As initial values, fix $=(0)=0\mathrm{.}0$ and $(0)=2($away from the potential in $).$ The range of sensible initial values for $(0),$ should be between $0\mathrm{.}9<=(0)<=0\mathrm{.}9$ but values around $0\mathrm{.}10\mathrm{.}2$ are quite interesting to observe. Likewise, pick an initial speed $0<=(0)<=0\mathrm{.}5.$

$($b$)$ Write a function scatterangles$($allb$,$ speed$)$ which takes an array of impact parameter from the interval $[0\mathrm{.}2,0\mathrm{.}2]$ with a step size of $0\mathrm{.}001$ as input as well as a speed value. Use the same time array as in part $($a$).$ Loop over the impact parameter values, solve the ODE's and determine the scatter angle. Again, as initial values, fix $=(0)=0\mathrm{.}0$ and $(0)=2.$ Fix the speed input value at $=0\mathrm{.}1$ when calling the scatterangles$($allb$,$ speed$)$ function before plotting the result, see below.

That scatter angle is determined from the final value of velocity components, i$.$e$.=$tan$1(/)$ at the maximum time value. Store the scatter angle at each loop iteration and return it as an array. Plot the scatter angle as a function of impact parameter with appropriate axes labels. Note that the imported python math function atan$2($y$,$x$)$ has the correct and safe return range of $<=<=$ since backscattering of the pinball can and will take place.