# Origin of the name ''momentum map''

Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?

• If the phase space is the cotangent of $R^n$ and the group is the group of translations, the moment map associates to every position-momentum covector $(q,p)$ the momentum $p$. If the phase space is the cotangent of $R^3$ and the group is the group of rotations, the momentum map associates to $(q,p)$ the angular momentum $p \times q$ (seen as a vector in the dual of the Lie algebra of $so(3)$). Jun 17 '16 at 12:52
• As far as I know the name was coined by Souriau, probably in: J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970. The reason for the name is explained by alvarezpaiva. The first translation into English was moment map, until Duistermaat pointed out that it should be momentum mapping. Jun 17 '16 at 16:11
• It's "moment map" on the east coast (of the US) and "momentum map" on the west coast. Sep 7 '16 at 9:25

According to §1.3 and §11.2 of Marsden and Ratiu  (see detailed citation given below), the momentum map concept can be traced back to Sophus Lie's 1890 book, and is an English translation of the French words application moment. To quote directly from Marsden and Ratiu:

The notion of the momentum map (the English translation of the French words “application moment”) also has roots going back to the work of Lie. Many authors use the words “moment map” for what we call the “momentum map.” In English, unlike French, one does not use the phrases “linear moment” or “angular moment of a particle,” and correspondingly, we prefer to use “momentum map.” We shall give some comments on the history of momentum maps in §11.2.

For additional clarification, I asked Tudor Ratiu about the origins of the term momentum map. In an email dated Sept 8th 2016, he gave a beautiful response on the evolution of ideas that led to the modern momentum map. At the very end, he made some comments on what Sophus Lie did. Here are the main points from his email, whose contents can mostly be found in the historical notes provided in Marsden and Ratiu . References are given at the bottom.

(1) The momentum map is a generalization of the usual notions of linear and angular momentum from physics, which have a long history.

(2) The first person to introduce the modern version of the momentum map was Bertram Kostant at a conference in Japan. He did not give it a name. He used it to prove what is today called Kostant's symplectic covering theorem.

(3) Just a few weeks later, Jean-Marie Souriau independently introduced the modern momentum map. He correctly named it in French application moment. He also realized its physical significance, and linked it to linear and angular momentum. This was an enormous breakthrough.

(4) Marsden and Weinstein, in their famous 1974 paper on reduction used the term moment map, which was their translation of the French words application moment. This had an immediate reaction from Hans Duistermaat who pointed out that this is a misnomer and physically incorrect. Indeed, even the wiki entry for moment, indicates in the first line that moment should not be confused with momentum. It's analogous to confusing force with linear momentum.

(5) For a circle action, the momentum map (without any name) was introduced by Theodore Frankel in an Annals paper in the 1950s. This paper is famous because it links the existence of fixed points of the action to the Hamiltonian character of the action. This bit of history was not known to Marsden and Ratiu when they wrote their book, so it won't be found there.

(6) Sophus Lie knew a lot of Poisson geometry and many people now regard him as the founder of Poisson geometry. However, Lie did not introduce symplectic geometry. The link to symplectic geometry, coadjoint orbits, etc. is due to Kostant and Souriau. See the historical notes in Marsden and Ratiu  for details.

References

Frankel, T. . Fixed points on Kahler manifolds. Ann. Math. 70, 1–8.

Marsden, Jerrold E. and Ratiu, Tudor . Introduction to Mechanics and Symmetry. Second Edition, 1999. Current 2nd Printing, 2003. New York, Springer-Verlag.

Kostant, B.  Orbits, symplectic structures and representation theory. Proc. US– Japan Seminar on Diff. Geom., Kyoto. Nippon Hyronsha, Tokyo 77.

Ratiu, Tudor. Personal Communication, Sept 8th, 2016.

Souriau, J.-M. . Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970. ISSN 0750-2435.

As alvarezpaiva mentioned, the name ''moment'' or ''momentum'' map comes from Physics and in particular from the Noether's theorem. Informally speaking, when there is a classical system that is invariant under rotations, then the corresponding Lie group representing this symmetry is $$SO(3)$$ and the physical quantity that is conserved is the ''angular momentum''. More precisely, the components of the map $$\mu: M \rightarrow \mathfrak{so}(3)$$, where $$M$$ denotes the phase space of the system, are conserved with respect to integral curves the Hamiltonian vector fields (i.e. the Noether's theorem).