## About The Project

This project was created for Computational Physics course (118094)
in The Department of Physics at
Technion - Israel Institute of Technology under the guidance of
Dr. Joan Adler.

In this project, a two dimensional
system containing one massive and
charged
classical
point particle which moves under the influence of
harmonic potential and a
constant in time and homogeneous in space
magnetic field perpendicular to the plane of motion will be
examined. The motion of the particle will be derived from
Newton's second law and from the
principle of least action using an
Hamiltonian. The attempt to solve this problem with
two different sets of equations is made to see if there is a fundamental performance difference in the solving process (because of reasons
discussed below).

Apart from studying the dynamics of the physical system, a comparison between two
numerical algorithms will be performed. One of the
algorithms to be compared is the the fourth-order Runge-Kutta algorithm, and the
second is a "Predictor-Corrector" algorithm which will be obtained by
combining the explicit Adams-Bashforth four-step method to make a prediction
and the Adams-Moulton three-step method to calculate the correction.

As there are two spatial dimensions in this problem, Newton's formalism produces
two ordinary differential equations (ODEs) of
second order which will then be
reduced in order to get
four ODEs of first order. In the Hamilton's formalism on the other hand,
although there are four ODEs of first order as well, the equation on the
conjugate
momentum of the angle is
homogeneous as the problem have an azimuthal symmetry -
the conjugate momentum is a constant of
motion. This leads to another check that will be made: what equations system is the more efficient to solve in terms of computation time and
numerical accuracy.

The program has two goals. The first is to make a comparison between the two numerical
algorithms in terms of time and
accuracy. This will be performed by comparing the numerical solutions to the
analytic solution which can be derived. The second goal is to produce
an AViz data files so the motion of the particle could be
visualized.

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