for starters think about the motion of a projectile with drag that is proportional to the square of velocity. Express the position in terms of cartesian coorinates and then transform to cylindrical. Comment on what set is better for a given set initial conditions.
To make things more complicated add invariable wind (or a constant applied force in an inertial direction)
Another complicated problem is to consider a particle attached to 2 springs. Each spring is connected to the ceiling and makes some angle between the horizontal and the particle. Each spring has a different spring constant. Find the equations of motion of the particle. Plot the phase between the x and y coordinates. Design spring 1 such that the particle will retrace its path with each cycle. Determine the maximum amplitide of the particles displacement from the equilibrium position. Assume m = 1 kg, theta1 = 30 deg, theta 2 = 60 deg, k2/k1 = sqrt(2). for the EOMs assume k1 = 1 N/m.
Derive the equations of motion for a spherical simple pendulum. (This is a pendulum that is free to rotate is both the horizontal and vertical planes)