How many equations of motion? The higher order derivatives are highly correlated Note: the bounty text above states "second order linear differential equations". It is an empirical observation that this is the case for the particular system I'm studying, please read "n-th order linear differential equation" instead
I have an object that moves in space, and I am trying to represent its movement in terms of differential equations.
This is the trajectory of the object:

This is a real-world object, whose position is sampled every thirty seconds. It moves slowly and accelerates slowly, so we will assume that the sampling rate is sufficiently fine-grained. Please just assume it: I can model the system computationally, later, and the $\Delta t$ can thus become arbitrarily small.
I want to learn a set of differential equations of the form $\frac{d^{n}x}{dt^{n}}$ for $n$ which varies between 1 and whatever it is the highest order derivative that adds information to this system.
Question: What is this $n$?
Here is my approach to answering this question:
These derivatives are computed by finite differences smoothed finite difference with a window frame of 5, and they look highly correlated with one another starting from order 3-4. The finite differences were discarded upon suggestion by @AndersSandberg , and now it appears that each derivative is correlated with its associated second order derivative:

This is the correlation matrix between each derivative and the next

I have around 90 objects that are following the same physics, and they all show similar pattern between the position and its time derivatives: from the pair $(\frac{d^1 x}{dt^1},\frac{d^3 x}{dt^3})$ onward, each additional derivative can be expressed as a linear function of the corresponding acceleration (twice-differentiation, $\frac{d^n x}{dt^n} = c_0 \frac{d^{n+2} x}{dt^{n+2}} + c_1$).
Is it correct to claim that, in this system, the equations of motion are these three:
$\begin{align*}
x(t) = c_{1,1} \times f_1(t) + 0 \times g_1 (x) + c_{1,3} \times h_1(\frac{dx}{dt}) + c_{1,4} \times i_1(\frac{d^{2}x}{dt^2})\\
\frac{dx}{dt}(t) = c_{2,1} \times f_2(t) + c_{2,2} \times g_2(x) + 0 \times h_2(\frac{dx}{dt}) + c_{2,4} \times i_2(\frac{d^{2}x}{dt^2})\\
\frac{d^2 x}{dt^2}(t) = c_{3,1} \times f_3(t)  + c_{3,2} \times g_3(x) + c_{3,3} \times h_3(\frac{dx}{dt}) + 0 \times i_3(\frac{d^{2}x}{dt^2})\\
\end{align*}$
The zero coefficients are for ease of read, the various $c$'s are some constants.
For each of the two coordinates of course. Please let me know if something is not clear, and thanks in advance!
 A: The plots indicate that higher order derivatives aren't providing any real additional information — and you probably cannot trust the 3rd and higher ones too much either, given the low granularity of your original data, regardless of the smoothing procedures used.
So it would be a mistake to focus on the relations between those curves or to try to infer too much from them.
As already suggested by Kyle Kanos in the comments, this motion definitely looks stochastic. You answered that, since your object is being attracted to a target, your system isn't stochastic, but notice that a system with both a deterministic and a stochastic terms (like virtually any real-world system, where noise is unavoidable) is called stochastic. This means that you'll probably be looking for a differential equation such as Langevin's, which includes a fluctuating random term, to model your object's motion.
As for the question of how many dimensions your system must have, you could try approaches such as time-delayed embedding or, if you insist in using the higher order derivatives, check how many can be dropped by applying dimensionality reduction methods.
