# Conservation of Linear Momentum with respect to a given direction

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.

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

## 2 Answers

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.

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.