I'm looking for simple problems in theoretical mechanics that are impossible or unreasonably difficult to solve by means of "brute-force" numerical integration of Newton or Euler-lagrange equations.
I'm interested in these beacuse I noticed that kind of "computer-nihilism" point of view is getting popular (at least among some students):
A person says, that "in the end we anyway doing real stuff by computer simulation. And for the numerical values of parameters we are usually able to numerically obtain the result with a given precision. So we just need to know how to write down the equations".
And, therefore, "there is no need to learn all that complicated stuff in theoretical mechanics".
Apart from obvious counter-arguments for this, I'd like to show that there are basic problems you are unable to solve without "the complicated stuff".
Let me give an example of such a problem:
Given:
- A center, that creates some strange field with the potential $U(r)=-\frac{\alpha}{r^3}$. (Mysterious planet)
- A body with mass $m$ scattering off this center. (Our space ship.)
- A radius R, at which we want to stay as long as possible.
Find: the impact parameter $\rho$ and the energy $E_0$ for our body, so it will stay in the "ring" $R<r<2R$ for as long as possible.
The problem is easily formulated. And it is easy to solve even for "newbies" in theoretical mechanics. The specific feature of the problem -- there is no reasonable way of solving it by doing straightforward computer simulation.
Can you propose other examples of problems with these properties?