In Landau Mechanics (third edition page 4), why does adding Lagrangians of two non interacting parts remove the indefiniteness of multiplying each Lagrangian by a different constant?
If both systems are completely non-interacting, I can perfectly create a new Lagrangian L = a*L1 + b*L2 and still maintain validity of the Euler-Lagrange equations and make the action integral perfectly stationary, with a and b different. Can't I?
Here's the original statement:
It is evident that the multiplication of the Lagrangian of a mechanical system by an arbitrary constant has no effect on the equations of motion. From this, it might seem, the following property of arbitrariness can be deduced: the Lagrangians of different isolated mechanical systems may be multiplied by different arbitrary constants. The additive property, however, removes this indefiniteness, since it admits only the simultaneous multiplication of the Lagrangians of all the systems by the same constant.
I kind of understand that, of course, after you create L = L1 + L2, xL = xL1 + xL2 but nothing stops me from multiplying each Lagrangian of each non interacting part by different constants a and b and then summing, like: xL = xaL1 + xbL2
So the statement seems without meaning for me. Could anyone clarify? Thanks!
One addition: I see that, if both systems are interacting externally, the constants must be equal (and the proportion of masses becomes relevant). My goal is exactly that: to understand, conceptually, how he derives the role of ration of masses later on in page 7. Going back to the text, what confused me is that the whole thing starts with the assumption that both system are closed so any external interaction is dismissed:
Let a mechanical system consist of two parts which, if closed (..), and the interaction between them may be neglected, the Lagrangian of the whole system tends to: lim L = La + Lb.