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Classical Physics
Mechanics
Effective potential in a central field
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[QUOTE="pbilous, post: 6310593, member: 674744"] Dear[USER=260864] vanhees71[/USER], thanks for your comment. I perfectly agree with you that the important difference between the energy and the Lagrangian in this context is that the energy is the first integral and is conserved, whereas the Lagrangian is changing somehow with the motion of the particle. However, I still don't see why it is necessary to restrict only to first integrals (your last sentence). I mean, we can express φ' via the angular momentum and substitute into the Lagrangian as well. Then we redistiribute the terms such that we get the "wrong" effective potential. The Lagrangian is not conserved, but I don't see in this fact any problem why we couldn't carry out the described procedure. On the other hand. Even if we stick to energy E = T + U = mr'[SUP]2[/SUP]/2 + M[SUP]2[/SUP]/(2mr[SUP]2[/SUP]) + U and introduce the "correct" effective potential Ueff = U + M[SUP]2[/SUP]/(2mr[SUP]2[/SUP]) (where M is the angular momentum), something strange happens to the Lagrangian. Indeed, the Lagrangian for the obtained effective 1D motion is L = mr'[SUP]2[/SUP]/2 - Ueff = mr'[SUP]2[/SUP]/2 - M[SUP]2[/SUP]/(2mr[SUP]2[/SUP]) - U. This differs from the initial Lagrangian by M[SUP]2[/SUP]/(mr[SUP]2[/SUP]), although the system described is exactly the same. This difference M[SUP]2[/SUP]/(mr[SUP]2[/SUP]) does not look like a full time-derivative, so these Lagrangians describe really two different systems. Do you have any idea how to resolve this contradiction? [/QUOTE]
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Effective potential in a central field
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