I have edited this question because I don't think that the related post answers my question fully. It refers to Noether's theorem but I would like an explicit illustration in an easier fashion: The angular momentum tensor is defined: $$L^{\mu\nu}~=~x^\mu p^\nu-x^\nu p^\mu$$
I would like to show that if a particle is not acted on by an external force that the angular momentum is conserved. Despite it being a direct consequence of the theorem I want to compute it in an inferior way, would this be sufficient?
Without loss of generality take the rest frame of the particle
$$\frac{dx^\mu}{d\tau}=(c,\underline{0})$$
Since the particle is not acted on by an external force, we have
$$\frac{dp^\mu}{d\tau}=\left(\frac{1}{c}\frac{dE}{ d\tau},\underline{0}\right)$$
$$L^{\mu\nu}L_{\mu\nu}=\left( 2 (x\cdot x) (p\cdot p) - 2 (x \cdot p)^2 \right)$$
Can I show the conservation by $$\frac{d(L^2)}{d\tau}=0$$
