Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1196

Nonlinear Dynamics of the 3D Pendulum

Nalin A. Chaturvedi·Taeyoung Lee·Melvin Leok·N. Harris McClamroch
View original source ↗Read source document ↗Export metadata ↓

Abstract

A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, invariant manifolds, local dynamics near equilibria and invariant manifolds, and the presence of chaotic motions. These results demonstrate the rich and complex dynamics of the 3D pendulum.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nalin A. Chaturvedi, Taeyoung Lee, Melvin Leok, N. Harris McClamroch. 2007-07-09. Nonlinear Dynamics of the 3D Pendulum. https://arxiv.org/abs/0707.1196

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lyapunov stability of polynomial vector fields is undecidable

We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.

math.DS↗
arXiv2026

On Standard perturbations of the Affine twist map

For any $t\in\mathbb{R}$, consider the Affine twist map $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$ given by $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ The map $\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of $\rm{Aff_t}$ to the vertical cylinder is reduced to $\{t\}.$ Now, for $k\in\mathbb{R},$ define $$f_{k,t}(x,y):=(x+y+k\sin (2πx)\text{ mod 1},y+k\sin (2πx)+t\text{ mod 1}).$$ We show that: 1) when $t=p/q$ for integers $p$ and $q>0$, KAM theory implies the existence of a constant $k_{p/q}>0$ such that for $|k|<k_{p/q}$, the vertical rotation set of an adequate lift of $f_{k,p/q}$ is just $\{p/q\}.$ 2) when $t$ is irrational, for any $k\neq 0$ the vertical rotation set of the adequate lift of $f_{k,t}$ is a non-degenerate interval which contains $t$ in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for $f_{k,t}$, for all non-zero values of $k$.

math.DS↗
arXiv2026

A topology for Engel expansions: evaluation and digit coding maps

We develop a topological framework for Engel expansions that treats both directions of the correspondence between points of $(0,1]$ and nondecreasing digit sequences. We endow the sequence space with the product topology to study the evaluation map, and we fix a nonterminating digit algorithm to study the digit coding map. We also record the correspondence between cylinder sets and fundamental intervals, and give an application to Baire category results for functions of the digits.

math.DS↗