# Invariance and forms of the Lagrangian

I have been doing Landau 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.

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The function can be one of velocity too if the initial and final velocities is also known or so I think. (This however causes the entire exercise to be useless since we shall know the entire trajectory). – Sourav Jul 3 '11 at 17:58
A comment about the question (v1): I think you will receive more precise answers if you told us which book of Landau, and in particular, which page for each sub-question. – Qmechanic Jul 4 '11 at 13:58
Landau's Mechanics (The 1st book of the series). I am talking mainly of the 1st chapter. – Sourav Jul 4 '11 at 18:41

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