Was the Lie bracket once called the Poisson bracket? (Milnor's Morse Theory)

Question

Multiple times in Milnor's Morse Theory, he refers to the "Poisson bracket" (or, once, an obvious typo: "poison bracket") of two vector fields as $$[X,Y](f)=X(Yf)-Y(Xf).$$ (See, e.g., the bottom of p. 4, or the line after Definition 8.5 on p. 48). Of course, this is not the Poisson bracket at all (which isn't even defined for two vector fields, as far as I'm aware), but is instead the Lie bracket.

I guess this isn't really a math question, but I was just wondering: Is there a reason why he keeps calling it the Poisson bracket? Was it historically called that, or were these just a couple typos on Milnor's part?

Answer 1

I think this is an interesting historical question; here are some comments (I believe a complete answer would require a more thorough investigation).

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Arnold also attributes the bracket commonly attributed to Lie to Poisson, see e.g. Mathematical Methods of Classical Mechanics, p. 208, Ordinary Differential Equations, p. 124, or Geometrical Methods in the Theory of Ordinary Differential Equations (2e), p. 182. This means that at least at some point in time some people understood by "Poisson bracket" what is typically called Lie bracket.

The first book I cited above (on the page I refer to) has a translator's note clarifying what Arnold calls the Poisson bracket (without reference to any symplectic structure) is also called the Lie bracket. The original Russian version seems to have been published in 1974 and the English translation in 1978. Milnor's book predates both by about ten years, and indeed Arnold refers to it; so it is likely that Milnor's attribution of the bracket in question to Poisson influenced Arnold, and at least by 1978 it was more common to make the attribution to Lie (at least in the English speaking mathematics community, according to the estimation of the translators).

I should also mention that my impression is that Milnor is rather accurate in such attributions (another instance is him attributing what is typically called the Frobenius theorem on the integrability of vector bundles also to Deahna and Clebsch; see the mimeographed notes of his titled "Foliations and Foliated Vector Bundles", p. 10). Of course this doesn't mean he is always accurate ($\ast$); still I tend to trust his attributions.

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From a chronological point of view it seems to me that it is also not completely unreasonable to speculate that attributing the bracket at hand to Poisson was common practice once. In this regard compare https://encyclopediaofmath.org/wiki/Poisson_brackets, https://encyclopediaofmath.org/wiki/Jacobi_brackets and https://encyclopediaofmath.org/wiki/Lie_bracket. This, together with the comparison matters of mathematics cultures in the States vs. in the Soviet Union (as well as biographical analyses of the people involved) require further investigation which I won't do. Instead I'll finish by pointing out Helgason's account of Lie's works (www-math.mit.edu/~helgason/sophus-lie.pdf) as a good starting point. There on p. 10 how Lie thought of Poisson bracket, among other things. I should note that in all the references in this section what is meant by the Poisson bracket is the Poisson bracket of functions w/r/t a symplectic structure.

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($\ast$) Added on 12/07/21: Case in point: On p. 4-17 of https://www.math.stonybrook.edu/~jack/DYNOTES/dn4a.pdf Milnor attributes a certain theorem (discussed in this other answer of mine here: https://math.stackexchange.com/a/4326733/169085) to Gottschalk & Hedlund, but Gottschalk & Hedlund themselves attribute said theorem to Schwartzman, in the book that Milnor refers to. Granted, this is not quite a publication, so it's not that big of a matter.

Answer 2

I don't know the history, but:

The bracket in a Gerstenhaber algebra is also called an odd Poisson bracket. On the supermanifold $\Pi T^*M$ with even coordinates $x^1,\ldots,x^n$ and odd coordinates $\xi_1,\ldots,\xi_n$ the Schouten bracket (odd Poisson bracket) takes the form $$[\![F,G]\!] = \sum_{i=1}^n\Biggl((F)\frac{\overleftarrow{\partial}}{\partial \xi^i}\cdot\frac{\overrightarrow{\partial}}{\partial x^i}(G) - (F)\frac{\overleftarrow{\partial}}{\partial x^i}\cdot\frac{\overrightarrow{\partial}}{\partial \xi_i}(G)\Biggr)$$ that also justifies the terminology.

The odd coordinates satisfy $\xi_j\xi_i = -(\xi_i\xi_j)$, and for the odd derivatives one has e.g. $$\frac{\overrightarrow{\partial}}{\partial \xi_1}(\xi_1\xi_2) = \xi_2 \qquad \text{ and }\qquad (\xi_1\xi_2)\frac{\overleftarrow{\partial}}{\partial \xi_1}=(-\xi_2\xi_1)\frac{\overleftarrow{\partial}}{\partial \xi_1} = -\xi_2.$$

Vector fields $\sum_{i=1}^n X^i \frac{\partial}{\partial x^i}$ on $M$ are identified with superfunctions $\sum_{i=1}^n X^i \xi_i$ linear in $\xi$.

Their Schouten bracket (odd Poisson bracket) $$[\![\sum_{i=1}^n X^i \xi_i, \sum_{j=1}^n Y^j \xi_j]\!] = \sum_{k=1}^n \biggl(\sum_{i=1}^n \biggl(X^i \frac{\partial}{\partial x^i}(Y^k) - Y^i\frac{\partial}{\partial x^i}(X^k)\biggr)\biggr)\xi_k$$ agrees with the Lie bracket of vector fields.

Bi-vector fields $\sum_{1\leqslant i<j \leqslant n}P^{ij}\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial x^j}$ are identified with superfunctions $P = \frac{1}{2}\sum_{i,j=1}^n P^{ij} \xi_i\xi_j$ quadratic in $\xi$. The Jacobi identity for a Poisson bi-vector field can be expressed as $\frac{1}{2}[\![P,P]\!] = 0$.