Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1108.4970

Action-angle coordinates for the pendulum problem

Alain J. Brizard
View original source ↗Read source document ↗Export metadata ↓

Abstract

The action-angle coordinates for the planar pendulum problem are expressed in terms of the Jacobi elliptic functions and integrals. In particular, we show that the Jacobi zeta function generates the canonical transformation from the pendulum coordinates $\vartheta$ and $p \equiv \partial\vartheta/\partial t$ to the action-angle coordinates $(J,ζ)$ for both the librating pendulum and the rotating pendulum.

Explore related subjects

physics.class-phmath-phmath.MP

Keep this discovery

BibTeXRIS

Alain J. Brizard. 2012-08-23. Action-angle coordinates for the pendulum problem. https://arxiv.org/abs/1108.4970

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

KEEP EXPLORING

Related papers

arXiv2026

Projection Angles of Projectiles in Sports: Qualitative Assessment of the Effects of Aerodynamic Forces or Run-Up

We examine two major factors that influence the optimum projection angle: aerodynamic forces and the effect of run-up. With respect to aerodynamics, we consider not only the drag but also the lift generated by spin during flight. By linearizing the equations of motion that include these forces, we derive perturbation solutions with respect to drag and lift coefficients and clarify their qualitative effects. The results show that both drag and lift reduce the optimum projection angle, with the latter exerting a stronger influence. To investigate the effect of run-up, we use an extended projection model in which the initial speed depends on the initial angle. Analysis of this model reveals that a stronger run-up increases the relative projection angle but decreases the launch angle observed from the ground. These findings provide a mechanical explanation for the release angle in shot put and the takeoff angle in long jump. The present study establishes a simple theoretical framework for clarifying the respective roles of aerodynamic and run-up effects in determining the optimum projection angles in sports.

physics.class-ph↗
arXiv2026

Dunkl-Based Modeling of Vibrational Modes in Lightweight Elastic Beams

Optimizing slender elastic structures for renewable energy applications requires non-classical continuum formulations capable of accounting for spatial micro-interactions without sacrificing analytical tractability. Here, we extend beam vibration mechanics by replacing standard spatial derivatives with the Dunkl differential operator. This modification introduces a reflection-coupled mathematical structure that accounts for spatial parity effects across the beam domain. We formulate the governing dynamic equations into a generalized eigenvalue problem and derive exact analytical expressions for modal characteristics under standard boundary conditions. The classical limit confirms exact convergence to classical Euler-Bernoulli formulations. Parametric analyses reveal that the Dunkl parameter acts as a reflection-induced modulation parameter, significantly shifting natural frequencies and altering the modal characteristics of higher modes. These results provide an analytical baseline for dynamic optimization in lightweight structural components.

physics.class-ph↗
arXiv2026

A purely mechanical system realizing a Coulomb-like interaction

We solve in closed form a one-dimensional relativistic system: two masses interacting only through elastic collisions with a massless mediator bouncing between them. Momenta, times, and positions are hyperbolic functions of the collision index. The mediator energy, interpreted as the pair's effective potential, obeys an exact discrete Coulomb law, $V\propto 1/r$, with a Lorentz-invariant action as coupling. A massive Newtonian mediator instead transmits a $1/r^{3}$ force; one adiabatic invariant traces both laws to the mediator's dispersion relation. Continued to negative mediator energy, the closed forms turn trigonometric, binding a one-dimensional mechanical analog of the Coulomb atom.

physics.class-ph↗