Looking for generalized formulas for Galilean transformations

[fisico30]

Dear Forum,

I am familiar with the formulas between inertial frames of reference that move at a constant speed between each other. The observed object move at a constant speed or at a constant acceleration. It can be shown that while the positions and velocities are different in the two frames of reference the accelerations are the same...

What if the two frames are accelerating with respect to each other? I have never seen the formulas...
What if the object are accelerating at a non constant rate? Have these situations be investigated analytically? I am sure they have but I have never seen a book describing these generalizations...

Thanks
fisico30

 

[VantagePoint72]

Yes, it's been looked at analytically; usually it's done with Lagrangian mechanics, but it can be done with Newtonian physics as well. If the two frames are accelerating with respect to each other, then you can describe the motion in the accelerated frame as being equivalent to an inertial frame with "fictitious" forces introduced.

So, if frame A is inertial and frame B accelerates linearly in the positive x direction, then motion in frame B will appear to be subject to a constant force in the -x direction. If B accelerates at a non-uniform rate, then that force will be time dependent (though the direction will stay the same).

If B is a rotating reference frame then the equations of motion in B will also contain a centrifugal force pushing objects away from the axis rotation, a Coriolis force pushing axially moving objects in the direction opposite to the rotation, and (if the rotation is non-uniform), an Euler force in the angular direction. When I say "pushing", I really mean "appearing to push" since in the inertial reference frame the objects just travel in a straight line. However, you could stick an observer in the non-inertial reference frame with a Newton metre and measure all these forces just like you would gravity or buoyancy, so in that sense fictitious forces are just as real as "real" forces. As a side note, in General Relativity the gravitational force essentially becomes a fictitious force too—so don't think they're unimportant just because of the name!

Here is a discussion on rotating reference frames:
https://en.wikipedia.org/wiki/Rotating_reference_frame

For linearly accelerating frames accelerating at $a$, you just need to an introduce an acceleration of $-a$ in the non-inertial frame and then use the usual kinematic equations.