Anyone read Landau's book on classical mechanics?
In section 7 he derives conservation of momentum from the lagrange equations by adding the assumption of homogeneity of space. He says that: "by virtue of this homogeneity, the mechanical properties of a closed system are unchanged by any parallel displacement of the entire system in space. Let us therefore consider an infinitesimal displacement epsilon, and obtain the condition for the Lagrangian to remain unchanged. A parallel displacement is a transformation in which every particle is moved by the same amount, the radius vector r becoming

Question

Anyone read Landau's book on classical mechanics? 
In section 7 he derives conservation of momentum from the lagrange equations by adding the assumption of homogeneity of space. He says that: "by virtue of this homogeneity, the mechanical properties of a closed system are unchanged by any parallel displacement of the entire system in space. Let us therefore consider an infinitesimal displacement epsilon, and obtain the condition for the Lagrangian to remain unchanged. A parallel displacement is a transformation in which every particle is moved by the same amount, the radius vector r becoming

anonymous · Answer

... (not enough space for the question) 

... the radius vector r becoming r + epsilon." 

Now, this is fine as long as we talk about a system consisting of one particle only, but if we talk about several particles not lying on a line, then such a displacement surely increases the distance between the particles and hence this cannot be a parallel displacement. Any opinions on this? Is this an error in the book, or do I miss something?

anonymous · Answer

sorry guys, I just realized that the epsilon was a vector and not a scalar, hence the definition makes sense after all