Simulation of a Newtonian and a relativistic plane-oscillator

Kostas Papamichalis Dr. of Theoretical Physics

The theoretical background and Exercises

The principal ojectives of this application is to get the user aquainted with the properties of the plane-relativistic oscillator and give him an appropriate environment to compaire the predictions of the Newtonian-Mechanics with the analogous predictions of the Special Theory of Relativity. The trajectory of a plane-Newtonian-oscillator is elliptic. This is not the case for a plane-relativistic oscillator: we shall see that the extrema of the length of the position vector are continually displaced.

The window of each simulation, with the attached orthogonal coordinate system (O,x,y,z) and the chronometer, constitute a Cartesian inertial system of reference. It must be noticed that all the chronometers placed at the space-points of this reference frame can be synchronized: they indicate the 'world time' (see the "Theoretical background").
The direction of the z-axis is to be considered perpendicular to the plane (O,x,y) of the simulation-frame.
The force-center is identical to the origine O. The motion of the particle is taken place on the plane (O,x,y).

The trajectory of the oscillating particle is determined by the following parameters, which are controlled by the user:
a) the initial velocity: (0,v0,0), and
b) the potential energy fraction: hp=V(r0)/mc2.
m is the mass of the oscillator and c the light-velocity. In the simulation the physical quantities are measured in atomic units. It is given that: c=1a.u., m=2000a.u.
The initial position of the oscillating particle, in both models, is (r0,0,0), where: r0=10a.u.

The position vector OA is controlled by the user (drag it by its end-point). He can use it to measure, on the virtual environment, the distance of the particle from O, and the angle formed by its position-vector with the OX-axis.

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Left simulation: The relativistic oscillator - Right simulation: The Newtonian oscillator

r0=10a.u.XYX'Y'OAx:undefined,y:undefinedtime t= 0.00angle(AOX)= 45.000deg, (OA)= 141.42aur0=10a.u.XYX'Y'OAx:undefined,y:undefinedtime t= 0.00angle(AOX)= 45.000deg, (OA)= 141.42au

The length of the position-vector versus the angle forming with the axis Ox

x:undefined,y:undefined0.090.0180.0270.0360.0450.0540.00.02.04.06.08.010.012.014.0angle u (degree)length r (atomic units)Relativistic oscillatorx:undefined,y:undefined0.090.0180.0270.0360.0450.0540.00.02.04.06.08.010.012.014.0angle u (degree)length r (atomic units)Newtonian oscillator


Title and author:

Relativistic Oscillator
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author image kostaspapamichalis