- #1

pbilous

- 2

- 1

^{2}/2 + L

^{2}/(2mr

^{2}). Finally we attribute the latter term to the potential energy and introduce the effective potential U

_{eff}= U + L

^{2}/(2mr

^{2}). The energy E remains the same under such term redistiribution and we obtain effectively 1D motion in potential U

_{eff}.

Now the question. If we consider the Lagrangian L = T - U instead of energy E =T + U and carry out the same procedure, then such "redistribution" of the terms would lead to different effective potential V

_{eff}= U - L

^{2}/(2mr

^{2}). Why is then the way with E "better" than the one with L?

Alternatively one could formulate it as follows. When we shift the term L

^{2}/(2mr

^{2}) to the potential energy U

_{eff}= U + L

^{2}/(2mr

^{2}), the total energy E = T + U remains the same, but at the same time the Lagrangian L = T - U changes by L

^{2}/(mr

^{2}). Doesn't it describe already a completely different system? Why wouldn't we choose to preserve L instead of E and introduce V

_{eff}= U - L

^{2}/(2mr

^{2})?