Use Python

Consider the dynamics of a metal strip hanging downwards from a fixed point, so that when straight its end is exactly midway between two strong

magnets. The strip of metal is flexible, and may bend towards one or the other magnet. If we assume there is no friction or heating in the system then the

energy of the metal strip is fixed and is given by:

$E=\frac{1}{2}{y}^2-\frac{1}{2}{x}^2+\frac{1}{4}{x}^4$

where $x$ is the position of the end of the metal strip between the magnets $($with $x=0$ exactly in the middle$)$ and $y=\frac{dx}{dt}$ is the velocity of the end of the

metal strip.

Import your libraries here:

a$)$ Write a function $E(x,y),$ that given a position $x$ and a velocity $y$ returns the associated energy $E$ of the metal strip. See if you can write your function so

that it receives a vector input $x=[x,y],$ this will make it possible to use the built in SciPy routine required in part c$).$

b$)$ Plot $E(x,0),$ i$.$e$.$ the plot of energy along the line $y=0$ in the range xin$(-2,2).$ What does this suggest will happen if the metal strip is initially

hanging motionless almost vertically?

c$)$ Find the position of the minimum for $x>0,$ and the value of the energy at the minimum.

d$)$ Plot the energy as a surface in interactive mode.

e$)$ Using the command $"$plt$.$ contour $()"$ create a contour plot of your energy surface. The contours in this plot are lines of constant energy. If energy is

conserved then the metal tip will be constrained to follow a contour. Identify which contour the tip will follow if it starts at rest slightly offset from the exact

mid$-$point between the magnets.