Exercise 3.1: Energy conservation

1. In Exercise 1.6 we wrote program to simulate simple harmonic motion. Now use the program to calculate the relative change in the total energy during one cycle $\Delta _n=(E_n-E_0)/E_0$. Is the function $\Delta$ uniformly small during the cycle? Choose $x_0=1$, $v_0=0$, and $\omega_0^2=9$ Compare using Euler's method and 2nd. order Runge-Kutta.

2. Plot position and velocity as a function of the time $t$.

3. Compute the amplitude $A$ for the initial conditions $x_0=4$, $v_0=0$, and $x_0=0$, $v_0=4$; choose $\omega _0^2=4$ in both cases. What quantity determines the value of $A$?.

4. Compute the average value of the kinetic energy and the potential energy during a complete cycle. Is there a relation between the two averages?

5. Plot the path of the oscillator in phase space $(x,v)$. Set $\omega_0^2=9$ and use different initial conditions $(x_0,v_0)=(1,0);(0,1);(4,0)$. Do you find different paths for each of them? What physical quantity distinguishes or characterizes each path? What is the shape of the phase paths? Is the motion of a representative point $(x,v)$ always in the clockwise or counterclockwise direction?