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In (Lawvere 97) it was observed that equations of motion in physics can (almost, see below) be formalized in synthetic differential geometry as follows.
Let $\mathbf{H}$ be an ambient synthetic differential topos (such as the Cahiers topos of smooth spaces and formal smooth manifolds).
The canonical line object $\mathbb{A}^1 = \mathbb{R}$ of this models the continuum line, the abstract worldline. Let
be the inclusion of the first order infinitesimal neighbourhood of the origin of $\mathbb{R}$ – in the internal logic this is $D = \{x \in \mathbb{R}| x^2 = 0\}$, externally it is the spectrum of the ring of dual numbers over $\mathbb{R}$.
Then for $X \in \mathbf{H}$ any object which we are going to think of as a configuration space of a physical system. For instance if the system is a particle propagating on a spacetime, then $X$ is that spacetime. Or $X$ may be the phase space of the system.
Accordingly the mapping space $[\mathbb{R}, X] \in \mathbf{H}$ is the smooth path space of $X$. This is the space of potential trajectories of the physical system.
If $X$ is thought of as phase space, then every point in there determines a unique trajectory starting at that point. This means that time evolution is then an action of $\mathbb{R}$ on $X$. As $X$ here might be any space, we have the collection
of all $\mathbb{R}$-actions on objects in $\mathbf{H}$. This is again a topos, and hence this is a first version of what one might call a topos of laws of motion.
On the other hand, if we think of $X$ as configuration space, then it is (in the simplest but common case of physical systems) a tangent vector in $X$ that determines a trajectory, hence a point in $[D,X]$. There is the canonical projection $[\mathbb{R},X] \longrightarrow [D,X]$ from the smooth path space to the tangent bundle, which sends each path to its tangent vector/derivative at $0 \in \mathbb{R}$. A section of this map is hence an assignment that sends each tangent vector to a trajectory which starts out with this tangent. Specifying such a section is hence part of what it means to have equations of motion in physics. Accordingly in Toposes of laws of motion Lawvere called the collection of such data a Galilian topos of laws of motion.
Of course this is not quite yet what is actually used and needed in physics. On p. 9 of (Lawvere 97) this problem is briefly mentioned:
> But what about actual dynamical systems in the spirit of Galileo, for example, second-order ODE’s? (Of course, the symplectic or Hamiltonian systems that are also much studied do address this question of states of Becoming versus locations of Being, but in a special way which it may not be possible to construe as a topos;
It turns out that it does exist as a “higher topos”.
In Classical field theory via Cohesive homotopy types it is observed that if $\mathbf{H}$ to be not just a topos but an infnity-topos, then genuine classical mechanics governed by Hamilton's equations and Hamilton-Jacobi theory arises by actions of $\mathbb{R}$ on objects in the slice (infinity,1)-topos $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$, where $\mathbf{B}U(1)_{conn} \in \mathbf{H}$ is the moduli stack of circle group-principal connections. So
is actually a “topos of laws of motion” in the sense of Hamilton-Lagrange-Jacobi classical mechanics. For more on this see Higher toposes of laws of motion.
The notion originates around
and was made more explicit in