Transformations that are scalar invariant

[junfan02]

I am a bit confused about something!
Exactly under what kind of transformations are scalars invariant in the domain of classical mechanics?
The fact which is disturbing me is, say we have a moving body of certain kinetic energy in a certain inertial frame of ref, and then we choose to.observe it from a different inertial frame moving with a certain velocity w.r.t the frst one. The kinetic emergy of the body is certainly not invariant?
So are scalars not invariant under translations and boosts( not.sure if it can be called so within the classical domain) ?

 

[Dale]

Hmm, I don't know if this will be helpful or not. There is the kind of "fuzzy" definition of a scalar as anything that isn't a vector, and there is the technical definition of a scalar as something that remains unchanged under any arbitrary coordinate transformation (diffeomorphism).

Energy, as you have indicated, is not technically a scalar. Technically, it is a component of the time-momentum four-vector. So it transforms as a component of a four-vector under arbitrary coordinate transformations.

Using the "fuzzy" definition you cannot say anything generally about the invariance of scalars under transformations. The "fuzzy" definition doesn't have fixed transformation properties so each one has to be taken on a case-by-case basis.

 

[junfan02]

Thank u very much!