Reading Landau's Mechanics, this question came in my mind. Suppose we have two non-interacting mechanical systems A and B, individually made up of a system of particles that are known to interact in some particular arbitrary manner. Landau states that, in the limit where the systems are separated by an infinite distance, the resultant Lagrangian tends to the value of sum of the individual Lagrangians.
This clearly implies that any potential that is due to a mutual interaction between two particles or two systems of particles must approach zero as the mutual distance is made to approach infinity.
From a dynamical systems theory viewpoint, could someone prove that this must necessarily imply that we get a globally defined flow in the phase space for any dynamical system following additivity of Lagrangians and least action principle. If not, can we perhaps say that these two ideas are not related?
//This is only a speculation, nothing concrete.