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Dynamics of a particle under the Gravitational Potential of a Massive Annulus: properties and equilibrium description

Eva Tresaco, Antonio Elipe, Andrés Riaguas

Abstract

This paper studies the main features of the dynamics around a massive annular disk. The first part addresses the difficulties finding an appropriated expression of the gravitational potential of a massive disk, which will be used to define the differential equations of motion of our dynamical system. The second part describes the main features of the dynamics with special attention to equilibrium of the system.

Dynamics of a particle under the Gravitational Potential of a Massive Annulus: properties and equilibrium description

Abstract

This paper studies the main features of the dynamics around a massive annular disk. The first part addresses the difficulties finding an appropriated expression of the gravitational potential of a massive disk, which will be used to define the differential equations of motion of our dynamical system. The second part describes the main features of the dynamics with special attention to equilibrium of the system.
Paper Structure (9 sections, 1 theorem, 52 equations, 9 figures)

This paper contains 9 sections, 1 theorem, 52 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The massive disk and its potential function
  3. Potential of a massive circular wire and disk
  4. Proper evaluation through potential reformulation
  5. Potential of a massive annular disk
  6. Dynamics around a circular annulus
  7. Dynamics on the $Oz$-axis
  8. Dynamics on the $xy$--plane
  9. Composition of annulus

Key Result

proposition thmcounterproposition

The dynamics of a particle under the attraction of a massive annulus and confined on the Equatorial plane verifies that

Figures (9)

  • Figure 1: Wire of radius $a$ acting on a point $P$
  • Figure 2: Disk of radius $a$ acting on a point $P$ with $r\leq a$
  • Figure 3: Disk of radius $a$ acting on a point $P$ with $r>a$
  • Figure 4: Left: Potential function $U = U(x,0,0)$. Right: Equipotential curves for the function $U(x,y,0)$
  • Figure 5: Phase portrait over the $Oz$ axis.
  • ...and 4 more figures

Theorems & Definitions (1)

  • proposition thmcounterproposition