Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Abstract

Escape from a potential well can occur in different physical systems, such as capsize of ships, resonance transitions in celestial mechanics, and dynamic snap-through of arches and shells, as well as molecular reconfigurations in chemical reactions. The criteria and routes of escape in one-degree of freedom systems has been well studied theoretically with reasonable agreement with experiment. The trajectory can only transit from the hilltop of the one-dimensional potential energy surface. The situation becomes more complicated when the system has higher degrees of freedom since it has multiple routes to escape through an equilibrium of saddle-type, specifically, an index-1 saddle. This paper summarizes the geometry of escape across a saddle in some widely known physical systems with two degrees of freedom and establishes the criteria of escape providing both a methodology and results under the conceptual framework known as tube dynamics. These problems are classified into two categories based on whether the saddle projection and focus projection in the symplectic eigenspace are coupled or not when damping and/or gyroscopic effects are considered. To simplify the process, only the linearized system around the saddle points are analyzed. We define a transition region, ${\mathcal{T}}_h$, as the region of initial conditions of a given initial energy $h$ which transit from one side of a saddle to the other. We find that in conservative systems, the boundary of the transition region, $\partial {\mathcal{T}}_h$, is a cylinder, while in dissipative systems, $\partial {\mathcal{T}}_h$ is an ellipsoid.

arXiv comment65 pages, 19 figures

References in corpus (15)

• The Time Scale of Escape from Star Clusters
• The formation of spiral arms and rings in barred galaxies
• Rings and spirals in barred galaxies. I Building blocks
• Rings and spirals in barred galaxies. III. Further comparisons and links to observations
• Manifold-driven spirals in N-body barred galaxy simulations
• Capture and escape in the elliptic restricted three-body problem
• Escapes in Hamiltonian systems with multiple exit channels: Part I
• A Tube Dynamics Perspective Governing Stability Transitions: An Example Based on Snap-through Buckling
• Isomerization dynamics of a buckled nanobeam
• Brouwer's problem on a heavy particle in a rotating vessel: wave propagation, ion traps, and rotor dynamics
• Orbital and escape dynamics in barred galaxies - II. The 3D system: Exploring the role of the normally hyperbolic invariant manifolds
• Experimental validation of phase space conduits of transition between potential wells
• Orbital and escape dynamics in barred galaxies - I. The 2D system
• An overview of the escape dynamics in the Henon-Heiles Hamiltonian system
• Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

Cited by in corpus (6)

• Geometry of transit orbits in the periodically-perturbed restricted three-body problem
• Saddle Transport and Chaos in the Double Pendulum
• Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems
• Painting the Phase Space of Dissipative Systems with Lagrangian Descriptors
• Phase space analysis of nonlinear wave propagation in a bistable mechanical metamaterial with a defect
• Transition criteria and phase space structures in a three degree of freedom system with dissipation