Third Newton's law as a DAG A we all know, Newton's third law is:
For every action, there is an equal and opposite reaction.
So, If A is pushed with force K by B, then it pushes back B with force k.
How can I represent this with a DAG?
I'm asking because I was reading Michael Nielsen's account of causality, and he says that "We can’t have X causing Y causing Z causing X! At least, not without a time machine. Because of this we constrain the graph to be a directed acyclic graph, meaning a (directed) graph which has no loops in it."
And I was wondering how we could fit this thinking with Newton's third law, since it seems that A causes B and B causes A at the same time, and thus we wouldn't be able to draw a DAG for this simple law of physics.
 A: The confusion in the causal concept here is based on a common misinterpretation of Newton's third law: that the force applied by A on B causes the equal and opposite force of B on A. This is not true. In fact, these forces happen simultaneously, and are both directly caused by external elements that dictate the force of each object on each other given by the laws of nature.
Consider the simple example of a binary system of stars with mass $m_A$ and $m_B$ separated by distance vector $x \equiv x_B - x_A$. We can write their equations of motion, which can be interpreted directly as causal equations:
$$
\begin{split}
\frac{d p_A}{dt} &= F_{\text{B on A}} = \frac{G m_A m_B\hat{x}}{\left|x\right|^2} \\
\frac{d p_B}{dt} &= F_{\text{A on B}} = \frac{-G m_A m_B \hat{x}}{\left|x\right|^2}
\end{split}
$$
where $p_A$ and $p_B$ are the momenta of stars $A$ and $B$ respectively, $G$ is the universal gravitational constant, $\hat{x}$ is the unit direction vector of $x$, defined earlier, and $F_{\text{B on A}}$ is the force that star $B$ exerts on star $A$.
The above equations can be interpreted directly as causal equations, where, in both cases, the masses and displacement vector on the RHS are causing the momenta on the LHS. It helps to understand that the forces happen simultaneously. One force is not causing another, but they are both caused by the masses of each individual object. This is true of all classical Newtonian interactions in which the third law applies.
The DAG would look like:

For ease of interpretation I added the force nodes, but in fact, it's equivalent to the terms on the RHS. So it would be just as meaningful to remove the force nodes and have the masses and separation causing the momentum nodes directly.

I should add, as an addendum, that this is how you would interpret things happening in classical Newtonian physics if you wanted to describe them in the language of statistics using Pearl's structural causal framework. However, in general, I think this is a bad idea. Physics and statistics mean very different things when they talk about "causality".
In physics, though causality means something different in classical Newtonian mechanics, quantum field theory, or general relativity, they all have a similar theme, in that you should look at it as the entire state of a system in one snapshot in time causing the entire future state of the system according to an evolution equation. (In relativity there's more restriction in what part of a state can affect other parts, based on limitations of the speed at which causal influence propagates.) If we take classical Newtonian mechanics as an example, the way in which we would talk about causality is this:


*

*Get all the information of a state, meaning every component's position and momentum, i.e. the "initial conditions"

*record the potential energy of the state, given by all its components and the manner in which your physics model tells you that components generate a potential energy

*calculate exactly how this state will evolve in the future. This evolution of the state was "caused" by the initial conditions


Essentially, in Newtonian mechanics, the entire state of the world is causing all future states of the world. This is a useful way to look at it if you're using your particular physics model to calculate "what happens" in a system, but it's not useful for the every-day way in which we talk about causality, which is more akin to the nodes and arrows we see in statistical causal frameworks. In the latter case, we isolate objects, call them A and B, and say A causes B. This latter way is more useful if we want to talk about intuitive sociological rules (e.g. poverty causes crime), but not when trying to see exactly how a system evolves in time, which is the goal of physics. For this reason, I think you should look at causal models in statistics as a separate thing from the notion of causality in physics. Think of it like different models that are valid for describing different size scales of the universe.
