\beesection{4.1}{The body bundle and the aerodynamic bundle}

\noindent
A bird in flight is the coupling of two fiber bundles.

\medskip

\noindent
\textbf{The body bundle.}\quad Let $\Sigma$ denote the topological 2-surface of the bird's skin. Topologically, $\Sigma$ is a sphere with a small number of cylindrical extensions---the wings, the legs, the beak. For our purposes it is a fixed smooth 2-manifold; the bird's identity does not change when its wings flex.

The \emph{body bundle} of the bird is
\[
  \pi_{\mathrm{body}} : \Sigma \times \mathbb{R}^3 \longrightarrow \Sigma,
\]
with base $\Sigma$, fiber $\mathbb{R}^3$, and projection forgetting the $\mathbb{R}^3$ factor.

A section of this bundle is a smooth map $s : \Sigma \to \Sigma \times \mathbb{R}^3$ with $\pi_{\mathrm{body}} \circ s = \mathrm{id}_\Sigma$, which is equivalent to a smooth map
\[
  \varphi_s : \Sigma \longrightarrow \mathbb{R}^3.
\]
If $\varphi_s$ is an immersion (injective on tangent spaces), the section $s$ describes a smooth placement of the bird's skin into three-dimensional Euclidean space. We call such a section an \emph{embedding section}.

\emph{The bird's instantaneous shape is an embedding section of its body bundle.} Wing extension, feather rotation, tail spread, neck flexion---all of these are operations that move the bird's current embedding section to a nearby embedding section within the space of all embeddings of $\Sigma$ into $\mathbb{R}^3$.

The space of embedding sections is infinite-dimensional. A bird has thousands of degrees of freedom available to it. Every individual feather can rotate; every joint can flex; the alula can deploy. A bird is, by any engineering measure, an obscenely high-dimensional control system that is solving an obscenely high-dimensional control problem in real time. We will come back to this in \S4.2 when we contrast it with what an airplane is allowed to do.

\medskip

\noindent
\textbf{The aerodynamic bundle.}\quad For a given embedding section $\varphi_s$, the bird occupies a specific region of three-dimensional space. The boundary of that region is the bird's body surface, embedded in $\mathbb{R}^3$. The surrounding air flows past this surface, and the flow is determined---at least classically---by the incompressible Navier--Stokes equations with no-slip boundary conditions on the surface.

The \emph{aerodynamic bundle} is, over each point $x \in \Sigma$, a finite-dimensional fiber consisting of the local aerodynamic state at that point of the embedded body: the local pressure $p$, the local viscous shear-stress vector $\boldsymbol{\tau}$ in the tangent plane, and the local relative airflow vector $\mathbf{u}_\infty$ near the surface.
\[
  \pi_{\mathrm{aero}} : \Sigma \times (\mathbb{R} \times T\Sigma \times \mathbb{R}^3) \longrightarrow \Sigma.
\]
A section of this bundle is a smooth assignment of $(p, \boldsymbol{\tau}, \mathbf{u}_\infty)$ to every point of the body surface.

\medskip

\noindent
\textbf{The coupling.}\quad An embedding section of the body bundle determines the boundary conditions of the Navier--Stokes problem in the surrounding air. The solution to Navier--Stokes determines an aerodynamic state at every point of the embedded body surface. This gives a map
\[
  \mathcal{N}:\; \Gamma_{\mathrm{body}} \;\longrightarrow\; \Gamma_{\mathrm{aero}},
\]
where $\Gamma_{\mathrm{body}}$ denotes the space of embedding sections of the body bundle and $\Gamma_{\mathrm{aero}}$ the space of sections of the aerodynamic bundle. The map $\mathcal{N}$ is nonlinear, non-local (the aerodynamic state at one surface point depends on the entire embedding), and---in any honest sense---only approximately computable. Computational fluid dynamics is the practical art of approximating $\mathcal{N}$.

\medskip

\noindent
\textbf{The wrench.}\quad The total force and moment on the bird is the integral of the aerodynamic state over the embedded body surface. A wrench has two components, a net force and a net moment about a chosen reference point. For an embedding section $\varphi_s$, with centre of mass at $\mathbf{r}_{\mathrm{cm}}$, and the induced aerodynamic-bundle section $\mathcal{N}(\varphi_s)$, the total wrench is
\[
  \boxed{\;
    \mathbf{W}(\varphi_s) \;=\; \bigl(\mathbf{F}(\varphi_s),\, \mathbf{M}(\varphi_s)\bigr),
  \;}
\]
with
\begin{align*}
  \mathbf{F}(\varphi_s) \;&=\; \oint_{\varphi_s(\Sigma)} \big[-p\,\mathbf{n} + \boldsymbol{\tau}\big] \, dA, \\
  \mathbf{M}(\varphi_s) \;&=\; \oint_{\varphi_s(\Sigma)} \bigl(\mathbf{r} - \mathbf{r}_{\mathrm{cm}}\bigr) \times \big[-p\,\mathbf{n} + \boldsymbol{\tau}\big] \, dA,
\end{align*}
where $\mathbf{n}$ is the outward normal to the embedded body surface, $dA$ is the surface area element, and $\mathbf{r}$ is the position of the surface element. The force integral $\mathbf{F}$ is the one most aerodynamics references write; the moment integral $\mathbf{M}$ is what makes the pair a wrench rather than just a force.\footnote{The sign convention here is the one in which $\mathbf{W}$ is the total force exerted \emph{on the body by the fluid}, computed with the outward surface normal. Equivalent formulations write the integrand as $\boldsymbol{\sigma}\cdot\mathbf{n}$, where $\boldsymbol{\sigma} = -p\,\mathbf{I} + \boldsymbol{\tau}$ is the full Cauchy stress tensor; some aerodynamics references use the inward normal and the opposite overall sign. Different conventions, same physical quantity.} The lift is the upward component of $\mathbf{W}$. The drag is the rearward component. Side-force is the lateral component. The moments about the bird's center of mass are the analogous integrals weighted by the lever-arm vector to each surface point.

\medskip

\noindent
\textbf{The control problem.}\quad The bird wants a particular total wrench $\mathbf{W}^*$: enough upward lift to support its weight, enough forward thrust to maintain airspeed, the right yaw moment to point toward where it is going. The control problem is: \emph{choose an embedding section $\varphi_s$ such that $\mathbf{W}(\varphi_s) = \mathbf{W}^*$.}

This is an inverse problem in an infinite-dimensional section space. The bird does not solve it analytically. The bird solves it by experience---hundreds of millions of years of evolutionary selection on the section-selection rule, plus the individual learning of every fledgling. The bird's nervous system computes a section-update from a small number of state observations (airspeed via feather-base mechanoreceptors, body-attitude via the vestibular system, local pressure via specialized skin organs) and a learned controller that is itself a section of a higher-order bundle.

The bird is a real-time inverse-problem solver in an infinite-dimensional section space. It is also a creature. Both of these are true. The first is what makes it possible to write a chapter about it. The second is what makes the chapter worth writing.

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