Ambiguity and a catalog of the actions

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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 S.

  1. The (off-shell) action

    (1)   \[S[q]~:=~ \int_{t_i}^{t_f}\! dt \ L(q(t),\dot{q}(t),t)\]

    is a functional of the full position curve/path q^i:[t_i,t_f] \to \mathbb{R} for all times t in the interval [t_i,t_f]. See also this question. (Here the words on-shell and off-shell refer to whether the equations of motion (eom) are satisfied or not.)

  2. If the variational problem (1) with well-posed boundary conditions, e.g. Dirichlet boundary conditions

    (2)   \[ q(t_i)~=~q_i\quad\text{and}\quad q(t_f)~=~q_i,\]

    has a unique extremal/classical path q_{\rm cl}^i:[t_i,t_f] \to \mathbb{R}, it makes sense to define an on-shell action

    (3)   \[ S(q_f;t_f;q_i,t_i) ~:=~ S[q_{\rm cl}],\]

    which is a function of the boundary values. See e.g. MTW Section 21.1.

  3. The Hamilton’s principal function S(q,\alpha, t) in Hamilton-Jacobi equation is a function of the position coordinates q^i, integration constants \alpha_i, and time t, see e.g. H. Goldstein, Classical Mechanics, chapter 10.
    The total time derivative

    (4)   \[ \frac{dS}{dt}~=~ \dot{q}^i \frac{\partial S}{\partial q^i}+ \frac{\partial S}{\partial t}\]

    is equal to the Lagrangian L on-shell, as explained here. As a consequence, the Hamilton’s principal function S(q,\alpha, t) 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.
  • Other researchers laboriously read his work and understand it.
  • Other researchers begin using the theory often and develop a condensed way of explaining it that sweeps details under the rug; notation is condensed to a form with significant ambiguities.
  • Much later, an author writes a textbook. He understands the theory well, but he uses condensed explanations and ambiguous notation because of habit, laziness or failure to appreciate the “curse of knowledge“.
  • Original researchers die. Original papers are buried in the avalanche of literature. Only a small number of researchers actually work on this theory, but everyone is expected to know the basics, i.e. it becomes part of the canon.
  • Everyone learns from textbooks, all of which use the same confusing explanations and notation.
  • Due to selection effects, it’s always possible to complete problem sets and pass courses in this theory using only the flawed understanding available in the textbook.
  • Student feel pangs of confusion, but they are conditioned to think the confusion is normal.Alternatively, the student (correctly) perceives that others will think they aren’t very smart if they repeatedly ask for clarifications in an attempt to alleviate confusion. The student wants to think of themselves as smart, so they decide they really do understand the theory.a  
  • Confused professors teach confused students. The cycle continues.

The only reliable way this cycle gets broken is when someone actually needs to understand what they’re doing (say, because it comes up in their research) and they have to either track down the old papers or derive it themselves from scratch.

Gerry Sussman of MIT wrote Structure and Interpretation of Classical Mechanics specifically to straighten this all out in classical mechanics. He takes an algorithmic approach, where all equations are written in machine-readable notation, so that all functional dependence is unambiguous. You can read about the philosophy behind this in the preface.

Indeed, there is a deep sense in which the different formulations of classical mechanics are characterized by their functional dependencies. In this context I can recommend the excellent short work Calculus of Variations by Gelfand and Fomin, which explains in the first 30 pages or so exactly what you need to know to understand how the calculus of variations is formally constructed without getting swamped by \epsilon‘s and \delta‘s. (After all, what is the point of learning the principle of least action if you couldn’t, in principle, explain exactly what it means to treat q and \dot{q} as independent variables?) See also Penrose’s The Road to Reality for beautiful illustrations of the geometric interpretations of partial derivatives.

Footnotes

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  1. Alternatively, the student (correctly) perceives that others will think they aren’t very smart if they repeatedly ask for clarifications in an attempt to alleviate confusion. The student wants to think of themselves as smart, so they decide they really do understand the theory.↵
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