# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

1,577 questions
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### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...
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### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
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### Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
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### Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
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### Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I ...
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### The “Dzhanibekov effect” - an exercise in mechanics or fiction? Explain mathematically a video from a space station

The question briefly: Can one explain the "Dzhanibekov effect" (see youtube videos from space station or comments below) on the basis of the standard rigid body dynamics using Euler's equations? (Or ...
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### Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...
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### Perron number distribution

A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any non-negative integer matrix $M$ ...
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### Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path ...
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### Finding a 1-form adapted to a smooth flow

Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
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### Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g., $$z^3+z^2+2 z+3 \;.$$ Find its $n$ roots, and list them in order of their modulus: $$-1.28, (0.14\pm 1.53 i)$$ Now form a new ...
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### 3D Billiards problem inside a torus

I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "...
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### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
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### Billiard dynamics under gravity

Has the dynamics of billiards in a polygon subject to gravity been studied? What I have in mind is something like this:            Still Snell's Law ...
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### Who first proved ergodicity of irrational rotations of the circle?

It is a classic result that the irrational rotations of the circle are ergodic. Formally, let $T:\mathbb{T}\to \mathbb{T}$ be defined by $Tz=ze^{2\pi i\alpha}$. If $\alpha$ is irrational, then $T$ is ...
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### Status of the 196 conjecture?

A palindrome is a number which remains the same when reversing it, for instance 34143. Now pick an arbitrary number, say 26: then 26+62=88 is a palindrome. If the number was 57, then 57+75=132 is not ...
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### A Collatz-like problem on prime numbers

Consider the function $f$ on the prime numbers defined by $$f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle (3,7,5,11,23,...
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### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...