__Amplitudes
and Periods of a Relativistic Oscillator
__

**Abstract.
**A**
r**elativistic oscillator is modeled on a computer. Two cases are considered:
(1) The mass is independent of the speed [m(v)=m_{0}] and (2) The mass
is a function of the speed [m(v)=m_{0}/(1-v^{2}/c^{2})^{1/2}].

We begin by noting
that the period of oscillation of a mass on a spring is *independent
of the mass’ maximum speed* when the mass is a constant of the motion (i.e.
when the mass always equals m_{0}).

In this article we
compute the motions of an oscillator when the maximum speed of the mass is a
relativistic 2.9e8 meters/second.
Computation for the two cases mentioned in the abstract are performed.

The oscillator in
each case can be considered itself to be a clock. Successive ticks can be
defined to be each time the mass passes through x=0 when traveling in the
positive x-direction.

The Python program
that computes the Amplitude and period of oscillation in each case is provided
in Appendix A. The results are as follows.

Case 1.
Amplitude=2.90e8 meters, Period=6.28 seconds.

Case 2.
Amplitude=7.23e8 meters, Period=11.36 seconds.

It is clear that both
the amplitudes and periods of the oscillator increase when the dependence of
mass on speed is factored in.

__Appendix
A
__

#Python
3.3.4 (v3.3.4:7ff62415e426, Feb 10 2014, 18:12:08) [MSC v.1600 32 bit (Intel)]
on win32

#Type
"copyright", "credits" or "license()" for more
information.

#This
program computes the period of a 'clock', which consists of a mass,

#attached
to an ideal spring of spring consfant k.

#The
mass oscillates back and forth on the end of the spring.

#The
mass' speed, at x=0 and t=0, is maximum and relativistic. Thus x=0 is the center
of

#oscillation.
The first of four legs of oscillation is assumed to end when the

#mass
speed is approximately zero. The goal of the modeling program is to compute the
amplitude

#and
the period of oscillation in two cases: (1) when the mass is independent of its
speed,

#and
(2) when the mass is a function of its speed.

import
math

n=10000000
#Loop counter

c=3e8
#Speed of light

v=2.9e8
#Maximum of mass speed (at x=0 and t=0)

m0=1.
#Rest mass

k=1.
#Spring constant

PeriodNonrel=2.*math.pi*math.sqrt(m0/k)
#Non-relativistic period

dt=PeriodNonrel/n
#Time between computation loops

x=0.
#Initial position of mass

gamma=1.
#Case 1. Choose this for non-relativistic mass

gamma=1/math.sqrt(1-v**2/c**2)
#Un-comment this for relativistic mass

f=-k*x
#Spring force on mass

a=f/(gamma**3*m0)
#Acceleration of mass

t=0.
#Time at beginning of oscillation

while
v>=0.: #Loop
for first quarter of oscillation

t=t+dt

x=x+v*dt+.5*a*dt**2

v=v+a*dt

gamma=1.

gamma=1/math.sqrt(1.-v**2/c**2)
#Uncomment this for relativistic mass.

f=-k*x

a=f/(gamma**3*m0)

Period=4*t

print('Amplitude=',x,'
Period=',Period)

print('End
of program')