In classical mechanics, is the curl of $\vec{v}$ always zero? As $\nabla$ is in position space and not in velocity space ($\nabla_v$). What am I missing regarding $\nabla$ operator in different spaces?
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4$\begingroup$ The curl of a velocity field is not always zero, otherwise all fluid dynamics would be about potential flow. But perhaps you need to be more specific about the problem you have in mind. If you are only considering particle dynamics in classical mechanics, velocity is not a vector field and you have no curl. $\endgroup$– kricheliCommented Sep 22, 2022 at 11:37
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1$\begingroup$ see this video for a visual understanding of the "curl of a vector field". Then, the fact that your vector field is a "velocity field" is not really important, you first need to grasp the general notions of "field" and "curl": youtube.com/watch?v=rB83DpBJQsE&ab_channel=3Blue1Brown $\endgroup$– QuilloCommented Sep 22, 2022 at 11:48
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You can only take the curl of a vector field, so $\vec{v}$ needs to depend on the position. This does happen in classical mechanics, typically in fluid mechanics. Then the nabla operator does act on position space (=contains derivatives with respect to position variables).
If you're studying the motion of a point-like particle, then velocity is a function of time only, so it isn't a vector field and its curl is undefined (or trivially zero, but it's a useless notion anyway).
