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 such that the vector space is equipped with the structure of a Lie algebra
and the bracket is ‘local’ in the following sense:
The most famous Jacobi manifolds are the Poisson manifolds, where the Lie bracket obeys this extra rule:
For any Jacobi manifold, the bracket can be written as
for some unique vector field and bivector field . and need to obey some identities to ensure that the bracket obeys the Jacobi identity. If we’ve got a Jacobi manifold with , so that
then our Jacobi manifold is a Poisson manifold, and is called the Poisson bivector or Poisson tensor. Conversely, any Poisson manifold has
for some bivector field .
So, generalizing from Poisson manifolds to Jacobi manifolds amounts to allowing a nonzero vector field in our formula for the bracket.
But Aïssa favors another way of thinking about Jacobi manifolds. Apparently a Jacobi manifold structure on gives a way of making some principal -bundle over into a Poisson manifold, where is the multiplicative group of the reals. The Poisson structures we get this way are homogeneous of degree with respect to the 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 bundle equipped with a Poisson bracket that’s homogeneous of degree ”.
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
http://www.northeastern.edu/tzhou/bhmn/Zhang19.html
that just happened in the big Boston colloquium. (I wasn’t there, just saw the abstract.)
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 or position , both can be seen as real-valued functions and thus there is no mathematical limitation to the definition of the observable . 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 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 , the first jet of the line bundle , where sections of the pull-back bundle , 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 is, locally, the contactization of the canonical symplectic structure of the cotangent bundle of the base: for some open . From this one sees that, morally, this formalism adds the freedom of choice of unit (the extra factor) to the geometric structure in a coherent way.
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 http://www.northeastern.edu/tzhou/bhmn/Zhang19.html that just happened in the big Boston colloquium. (I wasn’t there, just saw the abstract.)