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Is linear momentum conserved in any direction? More specifically, if you project all momentum vectors in a system onto another vector, will momentum be conserved?

I know that momentum is conserved along the $x$ and $y$ axes, so I'm expecting this to be true, but I have yet to see a rigorous proof.

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  • $\begingroup$ Comment to the question(v1): Are you considering an isolated system? $\endgroup$ – Qmechanic Oct 31 '12 at 21:53
  • $\begingroup$ yes, it is an isolated system $\endgroup$ – Ben Oct 31 '12 at 23:04
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Yes it is.

The total momentum vector of a system doesn't change at all (constant length and direction), so the projection of it on a line (or any function you apply to it) won't change. Projection is a linear operator, so that if you project each particle's momentum on a line and then sum, you get the same result as summing first (to get the total momentum) then projecting.

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In Euclidean geometry (as in classical mechanics), the axes are independent of each other; so yes, momentum is conserved along any vector. Really though -- when you factor in general relativity -- not so much.

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