In the following let a "mechanical system" be a system of $n$ spatial objects moving in physical space.
Consider you are given a function $q:\mathbb{R} \rightarrow \mathcal{M}^n$ with $\mathcal{M}$ a mathematical space.
Question 1
When would you assume that $q$ describes the evolution of a mechanical system?
I would, if I found or was given a function $\mathcal{L}(x,\dot{x},t)$ such that the Lagrange equations hold for all times $t$ when inserting $q(t)$ for $x$ and $\dot{q}(t)$ for $\dot{x}$, and which I can consider physically plausible. In this case I would interpret $\mathcal{M}$ as physical space, maybe distorted by generalized coordinates.
I would not, if there is provably no such function. In this case $q$ describes the evolution of a general dynamical system, but not a mechanical one.
Question 2
Are there mathematically definable conditions on $\mathcal{L}$ for being physically plausible, or possible, let's say?
Besides, maybe, that $\mathcal{L}$ must be of the form $\mathcal{T} - \mathcal{V}$.
Question 3
Can anyone give an example of a non-mechanical system that behaves like a mechanical one (in the narrow sense above)?
One answer to this question comes immediately in mind: a system that is designed to behave like a mechanical one, e.g. a computer running a simulation. I'd like to exclude such cases if I could, and restrict the question to "natural" systems.
Question 4
What about possible worlds with a physical space like ours but in which the Lagrange equations do not hold?