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.