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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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1 Answers1

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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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