Catastrophe theory

Convexity in 'native' variables may not always be so apparent. Change of variables, and expansion of dimension of the underlying space can lead to convexity. See for example the famous dynamic optimal transport : BENAMOU, Jean-David, and Yann BRENIER. "A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem." Numerische Mathematik 84.3 (2000): 375-393.

2 classics by VI Arnold: on classical mechanics and ODEs respectively.

Traffic models: helper.ipam.ucla.edu/publications/tratut/tratut_12985.pdf

Anecdotal answer: Kepler's equation is one of the most studied equations in the history of science. If there was a closed form solution, it would have been found by now. There are papers about its approximations appearing to this day!

Yes thats what i meant

Is the condition $\int f=0$ preserved by linearized dynamics ?

Are you familiar with en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem ?

I don't have a rigorous proof, but believe the above argument can be made rigorous without much pain.

Actually you may be right. The assumption can be removed since we are only concerned with finite-time invariance. No traj. from interior can escape to boundary in finite time, since that will violate backward-time uniqueness of the ODE, since the boundary is invariant.

I think the condition you need is simply $\langle v(t,x),n(x)\rangle < 0, \forall x\in\partial M$, where $n(x)$ is the outer normal, assuming the boundary is smooth.

That there is a wet lab in a US math department is probably a historical accident..but the world of fluid dynamics is full of applied math folks collaborating with physicists and engineers...so in itself it isn't unusual.

Good point, i take that back.

If the operator is not self-adjoint/symmetric, then the formula is for singular value, not eigenvalue.

Look up symplectic integrators

@user676464327 The history of mechanics is intertwined with math going back to Newton. Research in continuum mechanics, vibration, fluid mechanics, robotics all use sophisticated math ideas ..take your pick.

Easy answer: read physics texts. As a math undergrad, take courses in classical/quantum mechanics, fluid mechanics, relativity etc. Physicists have very different mental models for many familiar math objects, and it is a 'superpower' if you can have both.

The geometrical approach was initiated by Poincare , in ' les méthodes nouvelles de la mécanique céleste'.