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**What is "action"?**

I've seen various definitions of "action" in the context of a principle of least action. Lemons states it as the integral of momentum over path length. https://www.amazon.com/dp/0691026637/?tag=pfamazon01-20 Others, for example, John C. Baez, (http://math.ucr.edu/home/baez/classical/#lagrangian) present it as the time integral of the Lagrangian. He also gives an alternative definition, later in his presentation, which I have not read yet.

Are there different constructs known as "action" which cannot be shown to be identical, or are all of these definitions of "action" demonstrably equivalent?

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