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Usually in physics we use the notion generator to describe the infinitesimal elements associated with any finite Lie group transformation.

But in the context of the Hamiltonian formalism, all authors carefully use the notion generating function to describe the infinitesimal elements responsible for canonical transformations.

Is there a difference between the two concepts or can the word "generating function" be safely replaced with generator everywhere?

Concretely, the generating function responsible for a canonical transformation acts on the phase space coordinates via

\begin{align} q \to q' &= q + \epsilon Q \circ q = q + \epsilon \{Q,q\} \\ p \to p' &= p + \epsilon Q \circ p = p + \epsilon \{Q,p\} \, . \end{align} And usually, we call the objects responsible for infinitesimal transformations generators. I was wondering, why in the context of Hamiltonian mechanics all authors use the new term "generating function" instead. One guess would be that generating functions are the representation of our (abstract) generators in phase space.

Qmechanic
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jak
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    Presumably you mean this formalism? If so, note that it isn't limited to the description of infinitesimal transformations. – Emilio Pisanty Feb 20 '19 at 12:47
  • @EmilioPisanty thanks! I've edited the question to clarify what I have in mind. – jak Feb 20 '19 at 12:51
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    Well, it is a function on phase space that generates a transformation... so the natural thing to call it is a generating function. I'm not sure what exactly this question is asking. – knzhou Feb 20 '19 at 12:59
  • @knzhou I'm trying to understand if generating functions are elements of a Lie algebra, i.e. generators. – jak Feb 20 '19 at 13:03
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    In the formalism you are employing above, where $Q(q,p)$ is defined on points of the same phase space, of course they are generators of a Lie group flow, the Lie bracket being the PB, as reviewed here. In the alternate scheme linked by @Emilio Pisanty, they have a foot in each of the before-and-after phase spaces, and so a natural Lie algebraic structure is not readily apparent or practical, although I'd be shocked if it did not exit. They are functions. – Cosmas Zachos Feb 21 '19 at 02:28

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I found the answer on page 125 in Lagrangian and Hamiltonian Mechanics by Melvin G. Calkin.

A function $F$ is said to be a generating function because it allows us to calculate the new coordinates $Q,P$ from the old ones $q,p$ using \begin{align} p &=\frac{\partial F(q,P)}{\partial q} \\ Q &=\frac{\partial F(q,P)}{\partial p} \, . \end{align} The first equation here can be inverted to give $P(q,p)$. The second equation here gives us $Q(q,p)$.

In this sense $F$ "generates the transformation"

In contrast, the generator $G$ of the transformation describe infinitesimal canonical transformations. In general, we have

$$F_{inf}(q,P,t) = q P + \epsilon G(q,P,t) \, , $$ where $F_I(q,P,t) = q P$ generates the identity transformation. Using our equations from above we find for an infinitesimal canonical transformation \begin{align} Q &=\frac{\partial F(q,P)}{\partial P} = q + \epsilon\frac{\partial G(q,P,t)}{\partial P} \\ p &=\frac{\partial F(q,P)}{\partial q} = P + \epsilon\frac{\partial G(q,P,t)}{\partial q} \, . \end{align} Therefore, the infinitesimally transformed new coordinates read \begin{align} Q &=q+ \epsilon\frac{\partial G(q,P,t)}{\partial P}=q+ \epsilon\frac{\partial G(q,p,t)}{\partial p} \\ P &=p - \epsilon\frac{\partial G(q,P,t)}{\partial q} \, . \end{align}

Take note that above a type 2 generating function is used. The different types of generating functions are related by Legendre transforms. See, e.g. Tong's notes for further details.

mpv
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