I had to brush up on my Hamilton-Jacobi mechanics to referee a paper. I’d like to share, from this Physics.StackExchange answer, Qmechanic’ clear catalog of the conceptually distinct functions all called “the action” in classical mechanics, taking care to specify their functional dependence:

At least three different quantities in physics are customary called an action and denoted with the letter .

- The (off-shell) action
(1)

is a

functionalof the full position curve/path foralltimes in the interval . See also this question. (Here the wordson-shellandoff-shellrefer to whether the equations of motion (eom) are satisfied or not.)- If the variational problem with well-posed boundary conditions, e.g. Dirichlet boundary conditions
(2)

has a unique extremal/classical path , it makes sense to define an on-shell action

(3)

which is a

functionof the boundary values. See e.g. MTW Section 21.1.- The Hamilton’s principal function in Hamilton-Jacobi equation is a
functionof the position coordinates integration constants , and time , see e.g. H. Goldstein,Classical Mechanics, chapter 10.

The total time derivative(4)

is equal to the Lagrangian on-shell, as explained here. As a consequence, the Hamilton’s principal function can be interpreted as an action on-shell.

These sorts of distinctions are constantly swept under the rug in classical mechanics courses and textbooks (even good books like Goldstein). This leads to serious confusion on the part of the student and, more insidiously, *it leads the student to think that this sort of confusion is normal*. Ambiguity is baked into the notation! This is a special case of what I conjecture is a common phenomena in physics:

- Original researcher thinks deeply, discovers a theory, and writes it down.