(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We have a first frame S (named coordinate frame) at rest on the Earth (we suppose a non-rotating earht and this frame as a perfect inertial frame) and a second frame S' (named proper frame) moving in respect to the first with a constant relativistic velocity V along the x-axis.

The second frame is fixed to a relativistic rocket where onboard is placed a non-relativistic oscillator: mass-spring-damper system. The oscillator swings classically (that is at non relativistic velocities) and I want to write its equations of motion (and their solutions) in the proper frame and in the coordinate frame, also to find the tranformation of the spring constant and the damping factor from a frame to the other.

2. Relevant equations

I know the equation of motion in the proper frame:

m'*d2x'/dt'2+b'*dx'/dt'+k'*x'=0

where x' and t' are the coordinates in the S' frame, b' is the damping factor and k' is the spring constant in the same frame.

Besides I define:

gamma = 1/sqrt[1-(V/c)^2]

d2x'/dt'2 = a'

dx'/dt' = u'

3. The attempt at a solution

I can directly transform all quantities from this equation :

(m/gamma)*(a*gamma^3)+(b')*[(u-V)/(1-u*V/c^2)]+(k')*(x*gamma) = 0

but i cannot transform b' and k'.

Besides, being u' not relativistic can I do anyway this calculation?

thanks

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# Harmonic oscillator onboard a relativistic rocket

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