Question

# Is there a fundamental difference between coordination systems in classical, relativistic, and cosmological physics, what establishes a difference?

The choice of coordinate systems is a mathematical tool used to describe physical events. Local or universal spatial events occur in multiple coordinate systems of space and time or spacetime as we know it under classical, relativistic and cosmological physics.
Whether the fundamental laws of physics remains consistent across different coordinate systems.

Coordination system just imagination of any system. In that particular system, we just calculate any motion or any physical quantity in particular system. All systems do not contain infinite range, all theories are existing till particular coordinates. Dimensions could be change, but coordinate systems remain unchanged. So it doesn't matter.

The specific question, in my post, questions the application of Cartesian coordinate system in classical, relativistic and cosmological physics, and how the coordinate system is affected, if any of the fundamental laws of physics are inconsistent among classical, relativistic and cosmological physics,or not?
No. The choice of coordinate systems doesn't and can't matter. The property that can and does matter is the choice of the group of transformations that leaves the equations of motion invariant, whatever the coordinate system is chosen to be. The only effect the choice of the coordinate system can have is in simplifying the calculations.
In applications to cosmology, the group of transformations that leaves the equations invariant is the group of diffeomorphisms; in flat spacetime it's the group of Lorentz boosts, rotations and translations-the Poincaré group. A special case thereof is the non-relativistic approximation, when the group is the Galilean group.
It's remarkable that this isn't taught more forcefully. This hasn't been cutting edge research for at least a century...
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I thank you for your enlightening description which emphasizes in physics that the choice of coordinate system is not as important as the choice of transition group which preserves the fundamental equations of motion. These transformation groups ensure that the laws of physics remain consistent, regardless of the coordinate system chosen. Specific examples of these groups include variations in cosmology, Poincaré groups in flat spacetime, and Galilean groups in non-relativistic contexts. The primary effect of the choice of coordinate system is to simplify calculations, and these principles have been well established in physics for over a century.
These will definitely help me to proceed with my research work with more confidence. I thank you again for your enlightenment.
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Not only is the choice ``as important as the choice of the group of transformations'' but it doesn't make sense to discuss the two notions together.
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Yes. I thank you Sir Stam Nicolis .
Coordination system just imagination of any system. In that particular system, we just calculate any motion or any physical quantity in particular system. All systems do not contain infinite range, all theories are existing till particular coordinates. Dimensions could be change, but coordinate systems remain unchanged. So it doesn't matter.

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