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The Landau Lifshitz book "Mechanics" has a good reputation of one of the best books, not only on classical mechanics, but on theoretical physics in general. Yet, I have found a serious conceptual error (or at least sloppyness) in it.

Sec. 23 - Oscillations of systems with more than one degree of freedom:

In the paragraph after Eq. (28) they say that ##\omega^2## must be positive because otherwise energy would not be conserved. That's wrong. One can take negative ##k## and positive ##m##, which leads to negative ##\omega^2##, solve the equations explicitly, and check out that energy is conserved.

In the next paragraph they present a mathematical proof that ##\omega^2## is positive, but for that purpose they seem to implicitly assume that ##k## is positive, without explicitly saying it. A priori, there is no reason why ##k## should be positive (except that otherwise the solutions are not oscillations, but it has nothing to do with energy conservation).

Sec. 23 - Oscillations of systems with more than one degree of freedom:

In the paragraph after Eq. (28) they say that ##\omega^2## must be positive because otherwise energy would not be conserved. That's wrong. One can take negative ##k## and positive ##m##, which leads to negative ##\omega^2##, solve the equations explicitly, and check out that energy is conserved.

In the next paragraph they present a mathematical proof that ##\omega^2## is positive, but for that purpose they seem to implicitly assume that ##k## is positive, without explicitly saying it. A priori, there is no reason why ##k## should be positive (except that otherwise the solutions are not oscillations, but it has nothing to do with energy conservation).

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