Ordinary Differential Equations

1.

Write a program using Euler's Method and using the 2nd Order Runge-Kutta Method to find the phase space trajectory of a particle of unit mass in a 1-D potential

$$V(x)=(x^2-1)^2\,.$$

a.

Show the trajectory for the initial conditions

$$(x,p)=(1,0.1), (-1,0.1) \, {\rm and} \, (1., 10)$$

c.

Check the conservation of energy for different step sizes, for both the Euler Method and the 2nd Order Runge Kutta Method. Determine the step size where you have atleast $1\%$ accuracy in the energy over one whole period of the particle.

2.

Write a program to follow the motion of an electron in an electric field $\vec{E}(\vec{x},t)$ and a magnetic field $\vec{B}(\vec{x},t)$. Numerically determine the trajectory of an electron which starts at the origin with velocty $\vec{v}=(1,1,1) {\rm m/s}$ for the following field configurations

a.

Uniform magnetic field $10^{-4}$ Tesla along the z axis.

b.

Uniform magnetic field $10^{-6}$ Tesla along the z axis and an uniform electric field $1 {\rm V/m}$ along the $y$ axis.