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21 votes
0 answers
416 views

Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?

(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.) It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
1 vote
0 answers
108 views

Stability of rigid bodies spinning around $z$-axis under gravity

Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
164 votes
14 answers
40k views

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 "...
5 votes
3 answers
643 views

What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$?

Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$ and consider induced gradient-flow $$ \dot X(t) = -\nabla L(X(t)). $$ Question. Is ...
7 votes
1 answer
930 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations \begin{eqnarray*} \dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t)...
3 votes
0 answers
2k views

Bessel functions in wave propagation and scattering

Is there a way to scale $J_n(\cdot)$ (Bessel of first kind) and $H_n(\cdot)$ (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher values of n) and small arguments....
1 vote
0 answers
77 views

What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?

When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
10 votes
2 answers
350 views

Is this Riccati equation ("Josephson junction") always phase-locked at integer rotation numbers?

Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE $$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$ Identifying ...
101 votes
1 answer
8k views

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 ...
14 votes
2 answers
390 views

Is there a singularity theorem in higher-dimensional Newtonian gravity?

In classical Newtonian gravity with 3 spatial dimensions, it's hard to get two particles to exactly collide, since at short distance the centrifugal force (~1/$r^3$) beats the gravitational attraction ...
4 votes
0 answers
116 views

Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. ...
0 votes
1 answer
88 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
17 votes
5 answers
2k views

2- and 3-body problems when gravity is not inverse-square

Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing as $1/d^p$ for distance separation $d$ and some power $p$. Two questions: Presumably the 2-body ...