Isotropic of Inertial frame?

Question

My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and isotropic in space and isotropic in time. So it essentially imply that any physical quantity (say $L$) is same if it's at $r$ or $r'$ for a given $x$-$y$ inertial frame $S$ from homogeneity of space. Now it also implies that by isotropic of space, given any $r$ every direction gives same $L$ (and hence $L$ is a function of $\lvert\vec{v}\rvert$). I don't get how can we comment on isotropic of $L$ for a given $S$ because once we know $S$, we essentially know the origin $O$ and hence for a given $r$, the direction to look at it is fixed i.e radius vector $\vec{r}$.

P.S My question is closely inspired from my inability to understand the propertied of $L$ given in L.D Landau Mechanics on page 5.

Answer 1

Isotropy means that a theory has the rotations as a symmetry, homogeneity means that it has the translations as a symmetry. In natural language, isotropy is the requirement that the laws of physics look the same in all directions, and homogeneity is the requirement that they do not vary from point to point.

We say that the theory has a symmetry if the action $S = \int L \mathrm{d}t$ changes under the symmetry transformation by at most a boundary term since this means that the equations of motion do not change under the transformation, see this question and answer.

So, for the theory to be isotropic, the application of a rotation to our coordinate system must not change $S$. Since the time integral is unchanged under a rotation, this means that $L$ itself must not change under a rotation. The only thing we can associate to a vector $\vec r$ that's invariant under rotations and is not dependent on other vectors is its norm $\lvert \vec r \rvert$, so $L$ must be a function of $\lvert r \rvert$, but not of the vector $\vec r$ itself.