Use the Verlet method $($see script, section $5\mathrm{.}3\mathrm{.}1,$ from page $76-$ note that the expression F$/$m in that section is$,$ of course, the acceleration, the second order derivatives as shown below as ODE's$)$ to calculate the orbit of the Earth around the Sun. The equations of motion for the position $((),())$ of the planet in its orbital plane are:

$2()2=()3,2()2=()3,$

where $=2+2.$

Task: Calculate the orbit of the Earth around the Sun in steps of one hour for a total period of three years. Use the solution for the orbit, $(),$ and the velocity, $(),$ to calculate additionally the potential and kinetic energy of Earth at each evaluation point. Create a function potentialEnergy$($r$,$par$)$ and a function kineticEnergy$($v$,$par$)$ which implement these requested calculations separately from solving the ODE's and can be called from tests. Also create a function solve$($par$)$ which returns the value arrays for $()$ and $()$ as a tuple such that the tests can call solve$($par$)$ and obtain the results. Note that the potential energy depends only on the Earth location, $(),$ and its kinetic energy only on its velocity, $().$ The variable par should be a tuple containing the required constant values in order $($G$,$M$,$m$),$ see Data below.

Data: Use $=6\mathrm{.}6738\backslash$times $1011$ m $3$ kg $1$ s $2$ as Newton's gravitational constant, $=1\mathrm{.}9891\backslash$times $1030$ kg as mass of the Sun. For Earth mass, use $=5\mathrm{.}9722\backslash$times $1024$ kg$.$ The initial conditions should be $0=(1\mathrm{.}521\backslash$times $1011,0)$ m and $0=(0,2\mathrm{.}9291\backslash$times $104)$ ms $1.$ Time evaluation points are in the task description and note that you work in SI units.

Create two plots: $($a$)$ draw the kinetic energy, potential energy and total energy, superimposed on a single canvas and observe how the kinetic and potential energies change as a function of time while the total energy appears to be constant. Note that working in SI units means you will get large numbers of the order $1033$ J$.$

$($b$)$ Plot only the total energy as a function of time and observe the oscillating character while the mean remains constant to high precision which demonstrates the inherent energy conservation of the Verlet method.

Note: you may wish to solve the problem using solve$\_$ivp$()$ but that is not recommended. It is entirely possible and possibly illuminating in comparison$(!)$ to the Verlet method but effectively similar effort for worse results. The tests are likely to fail and deduct marks for this case.