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Isotropy of Space

Space is isotropic means one direction in space is identical or equivalent to any other direction. A particular experiment (i.e. physical process) will yield the same result whether our laboratory faces North or West. That is, an angular displacement of an isolated system does not change the internal state of the system, nor its internal motion.

When applied to an isolated system of two interacting particles, isotropy of space implies that the net work done by internal forces must be zero when the system is rotated by an angle dØ.

That is, d r₁ . F₂₁ + d r₂ . F₁₂ = 0

where d r₁ and d r₂ are displacement vectors of particles 1 and 2 respectively.

Space-Time Structure, Newton’s Laws and Gallilean Transformation

if r₁ and r₂ are constants.

Thus, we find

r₁ × F₂₁ + r₂ × F₁₂ = 0

that is, the net torque produced on the system by internal forces is zero. This implies that the angular momentum of the system remains constant. The above analysis can be extended to an n-particles system.

Thus, conservation of angular momentum of an isolated system emerges as a consequence of a fundamental property of space, viz. its isotropy.

Isotropy of space implies we can orient our co-ordinate axes in space in any way we wish.