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

## TL;DR

This paper explores the geometry and criteria of escape in two-degree-of-freedom systems with dissipative and gyroscopic forces, using tube dynamics to classify escape routes and analyze transition regions.

## Contribution

It provides a geometric framework for understanding escape in complex systems with two degrees of freedom, including effects of damping and gyroscopic forces, extending previous one-dimensional analyses.

## Key findings

- Transition region boundary is a cylinder in conservative systems.
- Transition region boundary becomes an ellipsoid in dissipative systems.
- Classifies escape routes based on coupling of saddle and focus projections.

## 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.