# Invariance and forms of the Lagrangian

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.

• 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

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).