I'm looking for any references for result relevant to the following conjecture:
Conjecture: For any computable Lagrangian $\mathbb{R}^{2n+1} \to \mathbb{R}$, $\mathcal{L}(t,{\bf q}, {\bf \dot{q}})$, then for any computable initial conditions ${\bf q}(0), {\bf \dot{q}}(0)$, ${\bf q}(t), {\bf \dot{q}}(t)$ are computable functions. Where computability is taken in the type-2 theory of effectivity sense as laid out by Weihrauch.
There is also the analogous conjecture for computable Hamiltonians and ${\bf q}, {\bf p}$.
In this physics stack exchange post, a user claim without apparent evidence that "classical mechanics is known to be non-computable", though I'm not really sure what that is intended to mean, also he didn't give a reference (and I haven't been able to find one, he reference "rapidly accelerating computer" as part of the proof of this claim, though that doesn't give any useful results on a first search). Of course we need constraints on what kinds of Lagrangians/Hamiltonians/force laws we are working with, because if these are themselves noncomputable functions, there are certainly cases where the particle trajectories are noncomputable.
Any relevant papers would be much appreciated. Of course, counterexample would be good as well. It's worth noting the there is a result by Pour-El and Richards that says the wave equation will time evolve "computable" initial data to a "non-computable" state for a computable time value. This is using I believe a different notion of computability from Weihrauch and there is a paper by Weihrauch and Zhong that explain that the Pour-El/Richards result doesn't violate the physical Church-Turing thesis and that the solution is always computable in the sense laid out in the paper.
PS: If any references for the case that we are working with field Lagrangians/Hamiltonians are known, they are also welcome.
Note: I have looked at Physics and Church–Turing Thesis on MO. I don't think anything there is useful to this problem.