I just read about Langrangian and Hamiltonian mechanics. I didn't encounter those at all in my EE physics, and they are fascinating. Great examples! Are you a physics professor, or is this stuff undergrad physics majors learn?
There's a good series of videos on YT, with the title Variational Calculus and the Euler-Lagrange equation on channel Structural Dynamics. I have only seen the first few. This first video should give you the full playlist:
That would only make sense to try if the model could do this for systems we already understand. By the sound of the article, it can't even do that. Despite many efforts the researchers couldn't even understand the second pair of parameters. That doesn't correspond to my understanding of "good parameters".
More importantly, though, they could use this NN on systems that have not yet successfully been modeled, perhaps complex dynamical systems, to discover good parameters and conserved quantities.