In my thesis I formalize composition of port-Hamiltonian systems in a *relational* way. That is, the dynamics of a port-Hamiltonian system are *related* to the input of the system, in a such a way that a given pattern of input could have none or many corresponding system behaviors.

Talking to @dspivak today I realized that there is another way of doing open classical mechanical systems which more directly divides things into input/output.

Unfortunately, after this week I won’t have time to work on this so much, so I want to do a quick write up of the story so far, so that I can either refer back to it in the farther future or so that someone else can be inspired.

The story begins with a version of Org that works for continuous-time dynamical systems.

I would be very surprised if @mattecapu hasn’t already described something like this, and @mattecapu if you are reading this please point me to a good reference if you have already come up with this.

Essentially, we start with the symmetric monoidal double category of lenses/charts between vector bundles on smooth spaces, apply the Para construction, and then look at the subcategory where the parameterizations are arenas of the form \begin{pmatrix} TX \\ X \end{pmatrix}.

Expanded a little bit, this is the double category where objects are pairs \begin{pmatrix} E \\ B \end{pmatrix} of a smooth space B (whatever your favorite definition of smooth space is, maybe just manifold) and vector bundle over it, vertical morphisms are charts, and a horizontal morphism from \begin{pmatrix} E \\ B \end{pmatrix} to \begin{pmatrix} E' \\ B' \end{pmatrix} is a space X along with a lens

These horizontal morphisms describe parameterized maps where the parameters smoothly change in response to information flowing backwards through the system. Call this double category \mathsf{COrg} (continuous Org).

Today with @dspivak, I came up with a way of constructing horizontal morphisms in \mathsf{COrg} which correspond to open classical mechanical systems.

I claim that given spaces A, X, B, a function f \colon A \times X \to B, a Poisson structure on X, and a function H \colon A \times X \to \mathbb{R}, I can construct a lens

The bottom map is simply f.

Now, for the top map, recall that a Poisson structure is given by a linear map J_x \colon T^\ast_x X \to T_x X that smoothly depends on x \in X, such that when viewed as a matrix J_x is *antisymmetric* (along with some other conditions).

Given a \in A, x \in X, and \phi \in T^\ast_{f(a,x)} B, we need to produce \psi \in T^\ast_a A and \dot{x} \in T_x X.

Recall that f^\ast_{a,x}(\phi), dH(a,x) \in T^\ast_{a,x} (A \times X) \cong T^\ast_a A \times T^\ast_x X, where f^\ast_{a,x}(\phi) is the pullback of \phi along f, and dH(a,x) is the gradient of H at (a,x). Then let

Now, I claim that this construction is in some way “natural”. What would it mean for this to be natural?

Construct a bicategory \mathsf{OpenCM} where the objects are spaces and a morphism from A to B consists of a Poisson manifold X, a function f \colon A \times X \to B, and a function H \colon A \times X \to \mathbb{R} called the Hamiltonian. Composition of (f \colon A \times X \to B, H_X \colon A \times X \to B) and (g \colon B \times Y \to C, H_Y \colon B \times Y \to C) is given by putting the natural Poisson structure on X \times Y, doing the normal Para construction to get a map (f ; g) \colon A \times X \times Y \to C, and then defining H_{X,Y} \colon A \times X \times Y \to \mathbb{R} by

The 2-cells in this bicategory should be symplectomorphisms X \to X' that preserve the Hamiltonian.

Then I hope that there is a symmetric monoidal bifunctor from \mathsf{OpenCM} to the horizontal bicategory of \mathsf{COrg} that corresponds to the construction that I did above. The reason that this is just a bifunctor and not a double functor is that sending A to \begin{pmatrix} T^\ast A \\ A \end{pmatrix} is not functorial when you want a chart from \begin{pmatrix} T^\ast A \\ A \end{pmatrix} to \begin{pmatrix} T^\ast B \\ B \end{pmatrix}, because we pullback cotangent vectors instead of pushing them forward. Perhaps with some galaxy brained @mattecapu stuff we can do a triple category of parameterized lenses, normal lenses, and charts, but I’m not there yet.

Future work includes:

- Proving that this is actually a bifunctor
- Relating this to John Baez’s Open Systems in Classical Mechanics
- Relating this to port-Hamiltonian systems
- Relating this to GENERIC systems