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February 4, 2019

Jacobi Manifolds

Posted by John Baez

Here at the conference Foundations of Geometric Structures of Information 2019, Aïssa Wade of Penn State gave a talk about Jacobi manifolds. She got my attention with these words: “Poisson geometry is a good framework for classical mechanics, while contact geometry is the right framework for classical thermodynamics. Jacobi manifolds are a natural bridge between these.”

So what’s a Jacobi manifold?

It’s really simple: a Jacobi manifold is a smooth manifold MM such that the vector space C (M)C^\infty(M) is equipped with the structure of a Lie algebra

{,}:C (M)×C (M)C (M) \{\cdot, \cdot \} \colon C^\infty(M) \times C^\infty(M) \to C^\infty(M)

and the bracket is ‘local’ in the following sense:

supp{f,g}suppfsuppg \mathrm{supp} \{f,g\} \subseteq \mathrm{supp} f \cap \mathrm{supp} g

The most famous Jacobi manifolds are the Poisson manifolds, where the Lie bracket obeys this extra rule:

{f,gh}={f,g}h+g{f,h} \{f,g h\} = \{f,g\} h + g \{f, h \}

For any Jacobi manifold, the bracket can be written as

{f,g}=(fdggdf)(v)+(dfdg)(Π) \{f, g \} = (f d g - g d f)(v) + (d f \wedge d g)(\Pi)

for some unique vector field vv and bivector field Π\Pi. vv and Π\Pi need to obey some identities to ensure that the bracket obeys the Jacobi identity. If we’ve got a Jacobi manifold with v=0v = 0, so that

{f,g}=(dfdg)(Π) \{f, g \} = (d f \wedge d g)(\Pi)

then our Jacobi manifold is a Poisson manifold, and Π\Pi is called the Poisson bivector or Poisson tensor. Conversely, any Poisson manifold has

{f,g}=(dfdg)(Π) \{f, g \} = (d f \wedge d g)(\Pi)

for some bivector field Π\Pi.

So, generalizing from Poisson manifolds to Jacobi manifolds amounts to allowing a nonzero vector field vv in our formula for the bracket.

But Aïssa favors another way of thinking about Jacobi manifolds. Apparently a Jacobi manifold structure on MM gives a way of making some principal *\mathbb{R}^\ast-bundle over MM into a Poisson manifold, where *\mathbb{R}^\ast is the multiplicative group of the reals. The Poisson structures we get this way are homogeneous of degree 1-1 with respect to the *\mathbb{R}^\ast action. I’m not sure I’ve got all the details right here, but she stated an equivalence of categories between Jacobi manifolds and something like “manifolds equipped with a principal *\mathbb{R}^\ast bundle equipped with a Poisson bracket that’s homogeneous of degree 1-1”.

This viewpoint sets up the connection to contact geometry. I understand how contact geometry is the right geometry for classical thermodynamics, and I understand how Poisson geometry is the right geometry for classical mechanics… so I should be almost ready to understand how Jacobi manifolds are a nice home for both these branches of physics!

But my insight quits right around here, so if you want more, I’m afraid you’ll have to try these papers, and the references therein:

By the way, Vitagliano and Wade are doing something a lot deeper than the simple stuff I’m discussing here. They are talking about holomorphic Jacobi manifolds, and answering the question “what is the global object whose infinitesimal counterpart is a holomorphic Jacobi manifold?” To understand this question you have to first realize that just as Lie algebras are infinitesimal versions of Lie groups, Poisson manifolds can be seen as infinitesimal versions of ‘symplectic groupoids’. This is a fascinating story! We can then generalize this idea to Jacobi manifolds… and then adapt it to holomorphic Jacobi manifolds. But I’m still interested in much more basic stuff, like: what are we doing if we use a Jacobi manifold rather than a Poisson manifold as the ‘phase space’ for a classical system?

Posted at February 4, 2019 7:40 PM UTC

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Re: Jacobi Manifolds

1) If one has a one-parameter family of noncommutative algebras, which at t=0 is commutative, by differentiating in t that commutative algebra gets a Poisson structure. When we extend this R^* action beyond t=0 to act on the whole family, should it scale t or leave it alone?

2) You’d probably have been interested in this talk that just happened in the big Boston colloquium. (I wasn’t there, just saw the abstract.)

Posted by: Allen Knutson on February 5, 2019 3:45 AM | Permalink | Reply to this

Re: Jacobi Manifolds

I spent lunch talking to Jun Zhang, who gave that talk in Boston! He’s here in Montpellier now — he regularly attends these Foundations of Geometric Structures of Information conferences.

I also had fun talking to Dmitry Alekseevsky, a guy from Moscow who knows an impressive amount about gradings of simple Lie algebras, symmetric spaces, manifolds with special holonomy, Jordan algebras and stuff like that. Today he’s talking about “The Vinberg theory of homogeneous convex cones, 66 years later”. I knew the theory of self-dual homogeneous convex cones, but not this broader theory.

Posted by: John Baez on February 5, 2019 6:37 AM | Permalink | Reply to this

Re: Jacobi Manifolds

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

Posted by: Eugene on February 7, 2019 3:19 PM | Permalink | Reply to this

Re: Jacobi Manifolds

Hi, I am a PhD student at the School of Mathematics in Edinburgh and I have been working precisely on the topic of using Jacobi manifolds as phase spaces for classical mechanics for some time now. There are several references on the use of contact geometry for time-dependent dynamics and dissipative systems, see for example:

This ties with your mention of the contact geometric formulation of classical theormodynamics.

Despite these interpretations being interesting and worthy of further exploration, I have found another avenue (to my knowledge, unexplored in the literature) where the use of Jacobi manifolds helps immensely in clarifying one of the conceptual issues that I have always found in geometric mechanics: the use of physical units of measurement (or lack thereof). My first research paper on the subject with title “Measurand Spaces and Dimensioned Hamiltonian Mechanics” is currently in preparation and should be ready soon (I will link it here when it is out).

The starting point of my investigations is the fact that practical physics, as in, experimental-test-ready mathematical models, always involve “dimensional consistency” of the physically meaningful quantities. This is often a physical intuition/ad hoc rule telling one to avoid the definition of physical quantities such as (Mass+Time) or (Length+Area), not a systematic or natural consequence of the mathematical axioms of the theory. This is particularly evident in the case of classical mechanics where observables are collectively considered as functions over phase space, thus forming a ring, to take advantage of the Lie algebra structure given by the Poisson bracket. For any typical mechanical system, examples physically meaningful observables are the kinetic energy KK or position xx, both can be seen as real-valued functions and thus there is no mathematical limitation to the definition of the observable K+xK+x. Physically we know that such quantity will never be part of any sensible theory but the mathematical structure seems to be blind to this fact.

In my paper I propose a slight generalization of Hamiltonian mechanics replacing configuration spaces and phase spaces as smooth manifolds with line bundles in such a way that observables are no longer real functions but sections of these line bundles. I explore the many ramifications of this change of perspective in mechanics (what happens to subsystems, mixed systems, conservation laws, reduction by symmetries, etc.) as well as give a motivation for such a generalization from an abstract review of the basic notions of metrology.

The punchline is that if one starts with a line bundle p:LQp:L\to Q as a configuration space (base manifold) with some information about the units of some physical quantity (1-dimensional fibres) it is possible to develop a formalism entirely analogous to canonical Hamiltonian mechanics by which one assigns the canonical contact manifold π:J 1LQ\pi:J^1L\to Q, the first jet of the line bundle LL, where sections of the pull-back bundle Γ(π *L)\Gamma(\pi^*L), what will play the role of observables, carry a natural Jacobi bracket. One sees these contact manifolds as the generalized phase spaces whose observables capture unit-free descriptions of physical quantities.

Note that the contact structure on J 1LJ^1L is, locally, the contactization of the canonical symplectic structure of the cotangent bundle of the base: J 1L| U=T *U×J^1L|_U=T^*U\times \mathbb{R} for some open UQU\subset Q. From this one sees that, morally, this formalism adds the freedom of choice of unit (the extra \mathbb{R} factor) to the geometric structure in a coherent way.

Posted by: Carlos Zapata-Carratala on February 14, 2019 7:22 PM | Permalink | Reply to this

Re: Jacobi Manifolds

Thanks! This question about the use of dimensional analysis in mathematical physics reminds me a lot of James Dolan’s ideas about ‘dimensional categories’ and algebraic geometry, explained here:

I’ll have to read the paper you linked to, but also the paper you’re writing, when you’re done with it.

Posted by: John Baez on February 14, 2019 8:14 PM | Permalink | Reply to this

Re: Jacobi Manifolds

Thank you very much for your reply! I was unaware of this notion of “Dimensional Category” but upon reading the nLab entry you linked, I realize that the category of Line bundles I use in my paper is indeed an example of this. In fact, I do a very similar discussion on the use of the category of Lines (1-dim real vector spaces) to model physical quantities as they are used in practical science. Then I move to the category of Line bundles to generalize Hamiltonian mechanics to a “unit-free” formulation.

I would like to know your opinion on a question about terminology: should it be dimensional or dimensioned? There are several references, mostly in more applied fields, dealing with “dimensioned numbers” or “dimensioned fields” as the mathematical models for physical quantities. I coined the term “Dimensioned Hamiltonian Mechanics” for my generalization based on this but if the name Dimensional Categories is really established then I think I would change it to “Dimensional Hamiltonian Mechanics”.

I will link the paper as soon as it is ready. Thanks again for your helpful comments!

Posted by: Carlos Zapata-Carratala on February 15, 2019 10:57 AM | Permalink | Reply to this

Re: Jacobi Manifolds

if the name Dimensional Categories is really established [….]

It’s not; this nLab article is the only place where it appears. James Dolan did a lot of research on this concept, some of which can be found on the nLab, but most of it is not written up. A very nice and closely related piece of work is here:

I don’t like the term “measurand” because it looks like a typo, but “dimensioned” seems fine.

Posted by: John Baez on February 15, 2019 6:47 PM | Permalink | Reply to this

Re: Jacobi Manifolds

Thanks for the answer and the reference.

I agree “measurand” is not the prettiest of words but my choice comes from two independent facts:

1) In the context of mathematics and theoretical physics, it is a compact 1-word term for a new concept (the line bundle generalization of observable) which doesn’t appear anywhere else in the literature.

2) In the context of metrology and applied science, “measurand” is quite an established term. From the International Vocabulary of Metrology:

“2.3 (2.6) Measurand: quantity intended to be measured. NOTE 1 The specification of a measurand requires knowledge of the kind of quantity, description of the state of the phenomenon, body, or substance carrying the quantity, including any relevant component, and the chemical entities involved.”

One of the goals of the paper was to link highly abstract mathematical physics (Jacobi geometry, category theory…) with practical science and engineering, so the term seemed ideal.

Posted by: Carlos Zapata-Carratala on February 16, 2019 1:23 PM | Permalink | Reply to this

Re: Jacobi Manifolds

Dear John,

I have now completed my thesis, it can be found here:

A Landscape of Hamiltonian Phase Spaces (thesis on ResearchGate)

A Landscape of Hamiltonian Phase Spaces (thesis on arXiv)

There you can find my current understanding of the topic of Jacobi manifolds as phase spaces in classical mechanics. I would recommend reading the conclusion (Chapter 9) for a quick overview of the work done withing the thesis.

Posted by: Carlos Zapata-Carratala on October 24, 2019 2:11 PM | Permalink | Reply to this

Re: Jacobi Manifolds

Thanks! I’m just coming back to this subject now; somehow I didn’t see your comment before.

Posted by: John Baez on December 27, 2022 10:50 AM | Permalink | Reply to this

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