The n-Category Café

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.)

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

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:

• M. de Leon and C. Sardon, Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems.

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 $K$ or position $x$, both can be seen as real-valued functions and thus there is no mathematical limitation to the definition of the observable $K+x$. 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 $p:L\to Q$ 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 $\pi:J^1L\to Q$, the first jet of the line bundle $L$, where sections of the pull-back bundle $\Gamma(\pi^*L)$, 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 $J^1L$ is, locally, the contactization of the canonical symplectic structure of the cotangent bundle of the base: $J^1L|_U=T^*U\times \mathbb{R}$ for some open $U\subset Q$. 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.