3 Springy pendulum [35%] In this question we will explore the spring pendulum system numerically using Python. This sys- tem consists of a mass on the end of a spring which is free to move in a plane. No small angle approximation is used in this problem so this system cannot, in general, be solved analytically. For certain parameters this system readily displays chaos, exploring this will be the focus of this question. This is the other kind of chaos that the lecture notes mention: Hamiltonian chaos. There is no forcing, no dissipation, energy is conserved and chaos takes the form of a complete exploration of the accessible phase space rather than collapses onto attractors. A schematic diagram of this system is shown in figure 2. Here I is the length the spring stretches to when the weighted system is in equilibrium, lo is the length the spring is when the unweighted spring is at equilibrium. Finally g and m are the gravity and mass respectively. A coordinate system for this setup can be defined a number of ways. In this assignment we will use an x-y coordinate system centred at the the equilibrium position of the mass. File Preview lo 00000000000 m X Figure 2: A schematic diagram of the spring pendulum system with the mass in the equilibrium position. The coordinate system and important variables are marked. 3.1. Show that the equations of motion for this system are k d x dt m x+ k m lox and (6a) d y dt == x + ( l y) - k lo (l-y) k -8+ (e y) m m 2x + (e y) (6b) Hint: It is useful to get expressions for sine and cost in terms of the length of the spring (note this is not, in general e) and use as an intermediary variable.