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gionole

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**Question 1:**in the non-inertial frame, space is non-isotropic. If we're in an accelerated train frame, and we face forward(the same direction where train is accelerating) and drop a ball, ball moves backward. If we face backward and repeat the experiment, dropped ball moves forward to us. So we got different results depending on where we were facing - hence non-isotropic, but I wonder, when we say physics don't behave the same in different directions(as It happened in our case), shouldn't the experiments yield the different equations of motions ? It seems to me that if our ##x## axis is in the same direction as train is accelerating to, in both of our experiments, the equation of motion is ##x(t) = -\frac{at^2}{2}##. Shouldn't it yield different equations of motion ? I'm asking this because it's important in the Lagrangian case where logic is that in the inertial frame, ##L## can't depend on direction, which is fine, but i wanted to bring counter argument that if we used non-inertial frame and it depended on direction, we would get the different equations of motions and break things, but turns out I get the same thing for my experiments.

**Question 2:**In the non-inertial frame, space is said to be non-homogeneous. Can you give a good example(not about rotating example), why space is non-homogeneous in such frame ? Mainly, what we do is we do experiment when object is at position ##x##, then repeat the experiment where object is at ##x+\delta x## and if physics doesn't change(equation of motion) stays the same, we say it's homogeneous. Would you be able to give an example in the accelerated frame, such as object's physics is different at ##x## and ##x+dx##(i.e equation of motion is different ? more like a really world use case example. Just very simple one where you don't come up with pendulum or anything. I tried to use just ball being in an accelerated train, but i don't get different results while dropping the ball at ##x## and ##x+dx##.

The reason why I mention non-inertial frame is non-isotropic and non-homogeneous, are shown in the attached image taken from Landau's book at page: 5

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