Why the existence of the potential function ##U## is not sufficient?

[Adesh]

Homework Statement

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Relevant Equations

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In Sommerfeld’s Lectures on Theoretical Physics, Vol II, Chapter 2, Section 6, Page 43 we derive an expression for the equilibrium of liquids as $$ grad ~p = \mathbf F$$ Where $p$ is the pressure and $F$ is the exertnal force. Then he writes,
[ The equation above ]includes a very remarkable Theorem: equilibrium is only possible if the external force has a potential, that is, if $\mathbf F$ can be represented as the gradient of a scalar function: $$ \mathbf F = -grad ~U$$ Where the minus sign is prompted by the relation to the potential energy. The existence of the potential function $U$ is not sufficient, $U$ must also be single valued within the space occupied by the liquid.

My problem is why existence of potential function is not sufficient? When he writes “$U$ must also be single valued” I couldn’t understand him as a scalar function will always be “single valued”. What he actually meant ? Please explain.

 

[Adesh]

I asked it to someone and he said that “single valued” means a kind of potential so that work done around a loop (in the field created by that potential) comes out to be zero. But I couldn’t understand what he meant, beacuse in the original text Sommerfeld wanted “$U$ to be single-valued” and he (Sommerfeld) said that $U$ was a scalar function so I don’t know what’s happening.