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John Taylor's Classical Mechanics says this... Image taken from John Taylor's Classical Mechanics

I was wondering if the second condition already implies the first? I mean, are there situations where the first condition is violated even though the second condition is not? And if so, how are the forces in that situation non-conservative even if they satisfy the second condition?

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  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/528336/2451 , physics.stackexchange.com/q/27896/2451 $\endgroup$
    – Qmechanic ♦
    Commented Sep 16, 2023 at 14:34
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    $\begingroup$ Does this answer your question? Why can't conservative forces depend on velocity? $\endgroup$
    – Bob D
    Commented Sep 16, 2023 at 15:01
  • $\begingroup$ I don't like the phrase "or any other variable" in condition one. The electric force depends on a particle's charge, in addition to its position, for example. $\endgroup$
    – Andrew
    Commented Sep 16, 2023 at 15:10
  • $\begingroup$ @BobD My question is more like, can there be force fields which satisfy the second condition without satisfying the first? And if this is not possible then the condition 1 is superfluous. As Adriano Del Vincio said in his answer that we can have time varying force fields which do not satisfy the first condition but they do satisfy the second condition at any fixed time.Considering a fixed time to check the second conditon is kinda logical otherwise consider this... the same path from A to B will lead to different work done on the particle depending on how fast the particle goes from A to B. $\endgroup$
    – user266637
    Commented Sep 17, 2023 at 7:19
  • $\begingroup$ @BobD So are there other examples? $\endgroup$
    – user266637
    Commented Sep 17, 2023 at 7:21

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I think that the first sentence has the purpose of avoiding those forces that are explicitly dependent on time or velocity. For particular forces that still depends on time the second sentece could be true (at a fixed time).

About the second sentence, I think it is tru that a conservative field can be derived from a scalar potential which depends only on the position r. The force is minus the gradient of the potential, so I think it is true that for a conservative field the second sentence implies the first. However one must also specifies that is time-independent and also speed-independent

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