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I am asking a conceptual question.

As we learned from classical mechanics, say Lagrangian formulation, as stated in Chap 7.9 of Classical Dynamics book by Thornton-Marion (5th Ed) p.260:

in our previous arguments, that time is homogeneous within an inertial reference frame.

So, we have "homogeneous time" within an inertial reference frame.

My conceptual question is that how do we formulate a relativistic classical mechanics, say Lagrangian formulation, such that the we have "homogeneous time" within an inertial reference frame, but that there are

time dilations and time contractions effects within special relativity classical mechanics?

In fact people have a successful Lagrangian formulation on the special relativity classical mechanics. But how does this work out conceptually? with this seemly dilemma:

"homogeneous time" v.s. time dilations and time contractions effects ?

p.s. I thought "Homogeneity" in time means the same "Homogeneity" as in fluid or the phase space. "Homogeneity" fluid means the fluid is non-contractible and non-compressible. But then I thought "Homogeneity" in time means the time is non-contractible and non-compressible.

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"Homogeneity" in time means being invariant under time translations.

Special relativity is invariant under the Poincaré group, and so in particular time translations.

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