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  1. Will Linear momentum be conserved in a non-inertial frame of reference?

  2. In other words what is the fundamental condition for linear momentum to be conserved?

  3. Also which is more fundamental- Newton's Third Law or Conservation of Linear Momentum? (as in Newton's third law does not hold true when we go to the quantum/atomic level or at relativistic speeds whereas linear momentum to my knowledge holds true)

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Linear momentum will be conserved when the Lagrangian that describes your system is unchanged by translations in space. This is a consequence of Noether's theorem, and it's as close as you're going to get to a fundamental explanation for the conservation of momentum.

In general a Lagrangian written in the coordinate system of an accelerating observer is not going to satisfy this condition, so momentum will not be conserved.

I suspect most physicists would regard Newton's third law as derived from conservation of momentum.

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