Invariance and forms of the Lagrangian

Question

I have been reading the 1st chapter of Landau & Lifshitz Mechanics, and due to its concise style been facing a few problems. I hope you can help me out here somehow.

1. Does the "homogeneity of space and time" essentially talk about the invariance of the Lagrangian with co-ordinate system or time invariance? Is this the basis of Landau's proof of the Law of Inertia?

2. The Lagrangian for a closed system of particles is described by adding a function of the co-ordinates. This essentially means that the same trajectory satisfies the extremal principle. (I am a bit confused about this though). So why the function added is not of time too?

3. On the subject of " the Lagrangian being defined only to within an additive total time derivative of any function of co-ordinates and time", is it true that the function can be of velocity too (Though this would make the entire exercise useless, since we would know everything)?

The disadvantage of trying to teach oneself physics without anyone to refer to is that I am not completely sure that what I am saying makes sense.

Answer 1

1. Yes it is about invariance with respect to time and space translations: $t\to t+t_0$ and $\vec{r}\to\vec{r}+\vec{r}_0$

2. I see no fundamental problen in introducing the time dependence of the potential energy. But it would mean that the way our particles interact with each other changes with time somehow. Seems to me that it is pressuposed that the interaction cannot change in closed systems -- rather natural assumption for me...

3. The problem with a function of speed is that the new "lagrangian" will depend on second derivative of coordinates: $$L'(q,\dot{q},\ddot{q},t)=L(q,\dot{q},t)+\frac{d}{dt}f(q,\dot{q},t)$$ Which contradicts to the experimentally established statement that a system fully described by coordinates and velocities (begining of chap.1).

... trying to teach oneself...

I'd stronlgy discourage to study mechanics by means of Landau only. He sometimes is too fast -- there are some subtleties that he skips without paying attention. Also in some places he uses non-standard terminology, which can be misleading when dealing with other sources.
I'd recommend Mathematical Methods of Classical Mechanics by V.I. Arnold. Personally I find it to be very nice complement to Landau.

Answer 2

The answer of Kostya gives the insights very well. However I would like to add that your first and second points are quite related. When you assume that time is homogenous you find the law of the conservation of energy. Now if the Lagrangian depended directly on time the Energy Conservation would be violated. A simple example is when there is an external field dependig on time.