Mechanics Landau Galilean Principle

Question

I started reading Landau's Mechanics book and was having some trouble understanding the Galilean Relativity Principle. What does Landau mean by saying space to be homogenous and isotropic and time is homogenous for an inertial observer? Also, how does he conclude using this that, the Lagrangian must be independent of $\vec{r}$, time and direction of $\vec{v}$ .

I am not understand what he wants to convey.

Also a side note, are there any lecture notes/books/videos etc. that explain in a bit more detail what Landau wishes to convey to the reader at times in his book.

Answer 1

What does Landau mean by saying space to be homogenous and isotropic and time is homogenous for an inertial observer ?

"Homogeneous space" means that all points of space are physically equivalent. He then uses this as motivation for the requirement that the Lagrangian function of a system in such homogeneous space should not depend on where exactly the system is placed. This then implies that the total momentum

$$ \sum_k \frac{\partial L}{\partial \dot{\mathbf r}_k } $$ is conserved.

Similarly, "homogeneity of time" means that all time instants are equivalent. He then uses this to motivate the requirement that the Lagrangian function of the system should be independent of the time coordinate $t$ (beware, actually the value of the Lagrangian will depend on time, even if the Lagrangian function $L(\mathbf r,\dot{\mathbf r}) $ does not depend on it). This then implies that the quantity

$$ \sum_k \mathbf{p}_k\cdot \dot{\mathbf r}_k - L $$ is independent of time too. This is often equal to familiar expression for the total energy of the system.

If you're looking for a more digestible first reading, try lectures notes Classical Dynamics by David Tong: http://www.damtp.cam.ac.uk/user/tong/teaching.html

Answer 2

In this context, the fact that the space is homogeneous means what follows.

Assume to have an isolated mechanical system. If I fix an initial condition (position and velocity of every point of the physical system at the initial instant of time), I find a corresponding evolution solving the equation of motion. What happens to the resulting motion of the system if I change that initial condition (at the same initial time) by means of given translation $T$?

In inertial reference frames the answer is simple even if by no means physically obvious:

The resulting motion is exactly the previous one, with the only difference that at each instant of time every reached configuration has to be changed by the action of that initial $T$.

If it happens for every mechanical system and every translation, the rest space of the used reference system is said to be homogeneous. This is the case for inertial frames, but this property generally fails for non-inertial reference frames.

Replacing translation for rotation, the notion of isotropic space arises. Replacing rotation with time translation, one has the notion of homogeneous time. There is a remaining notion due to pure Galilean invariance, i.e. $t\to t$, ${\bf x} \to {\bf x} + t {\bf v}$ for all possible vectors ${\bf v}$. Dynamics in inertial frames is also invariant under that class of transformations.

(All that is related with the 10 one-parameter subgroups of Galileo's Lie group).

Coming to the Lagrangian formulation, the Lagrangian of a point can, in principle, be a function of time, position and velocity of the particle.

It is not so difficult to see that, in view of the structure of Eulero-Lagrange equations, imposing (1) homogeneity, (2) isotropy, (3) time homogeneity and (4) invariance under pure Galilean transformations for a point of matter, its Lagrangian has to assume the standard form $k \dot{x}^2$. (I am not sure whether or not the invariance under pure Galilean transformations can be dropped as an hypotheses.)