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a spring-mass-damper (viscous damping) parallel to the flight direction (we suppose to have the system attached to a relativistic spacecraft). i would like to study the case with an external force (constant) and without it.

we have a proper frame (attached to the spacecraft and the system) moving with a constant relativistic velocity V, and a coordinate frame at rest on the earth.

i would like to write the motion equations of the mechanical system in the proper frame and in the coordinate frame also in order to yield the spring constant and the damping factor in both of the frames.

someone said me to write the lagrangian in both of the frames (but i'm not able to do it correctly) and someone else said to write directly the motion equations in the proper frame and transform all quantities to the coordinate frame. but in this second way, how can i do to transform the sprign constant? and the damping factor?

besides, if te mechanical system is perpendicular to the flight direction?

in both of the cases, only V is relativistic, the mass motion (in the proper frame) is very slower.

thank you very much for your help

Luca