A small puck oscullates in a dip near the top of a hill under the influence of gravity (you can neglect friction). The height of the hill as a function of a horizontal coordinate a is given by y = H + ax br and you can assume that the amplitude of the oscillations is small enough so that bamax/a 1 (and rmax is the amplitude oscillations). (a) Write down Lagrangian of the system. (b) Solve equations of motion for the system to the leading order in small parameter c. Write the solution in the form r(t) = ro(t) + x1(t) where ro(t) is the zeroth order approximation and r1(t) is the first order correction. Write the frequency as w? = w3 + Awi, where wo is the frequency of oscillations in the absence of at term and Aw? is an anharmonic correction to the frequency. Find w in terms of g and a. Find Awf to leading order in the small parameter b.