Differential Equations and Dynamical Systems

What Hacker News says

1. kxyvrJan 2023
   1. Principles of Mathematical Analysis by Walter Rudin (baby Rudin) - I'd studied real analysis in the past, but this book is direct and rigorous and provided a good framework to move forward into things like functional analysis in a way that I was not prepared for with other books.

   2. Differential Equations and Dynamical Systems by Lawrence Perko - Solidified for me how dynamic systems behaved and were solved. Very much helped my understanding of control theory as well.

   3. A Concise Introduction to the Theory of Integration by Daniel Stroock - Helped solidify concepts related to Lebesgue integration and a rigorous formulation of the divergence theorem in high dimensions.

   4. Convex Functional Analysis by Kurdilla and Zabarankin - Filled in a lot of random holes missing in my functional analysis knowledge. Provides a rigorous formulation of when an optimization formulation contains an infimum and whether it can be attained. Prior to this point, I often conflated the two.

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2. throwaway81523Jun 2022
   I liked Hirsch and Smale's old book called something like "Linear algebra, differential equations, and dynamical systems". It is now replaced by an expanded edition with a 3rd author added and a longer title, that I expect is also good, though I haven't looked at it.

   I don't know if the H&S book is beginner friendly, but what I found good about it was studying linear algebra and differential equations at the same time, i.e. treating them as closely related topics rather than separate ones. So you could use your physical intuition about (say) a harmonic oscillator (mass on a spring, the archetypal second order ODE), then see how the 2nd order equation can be separated into a system of first order ODE's, and solved by finding matrix eigenvalues.

   That worked better for me than the abstract linear algebra approach that was purely about vector spaces with nothing going on in them. It showed real sensible motivations of linear algebra.

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3. throwaway81523Aug 2021
   Yes we have a proof (in the classical case). It is called the Picard–Lindelöf theorem. The "dome" in your example is called an unstable equilibrium: the ball starts rolling because of an unpredictable perturbation. Small motions are amplified according to the Lyapunov exponent, which in the unstable case is large. Hirsch and Smale and Devaney's book on differential equations and dynamical systems is a good place to learn about this.

   In classical mechanics we model a zero-diameter point at the top of the dome, but a real physical dome would be made of molecules vibrating randomly and that starts the ball rolling. At the end of the day, classical mechanics is nonphysical: it doesn't describe nature except as an approximation. Quantum mechanics on the other hand is non-deterministic. Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell.

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