About the intuitive picture of a Hamiltonian for a dissipative system

TL;DR

This paper presents a Hamiltonian formulation for a 1-D dissipative system, revealing contrasting behaviors of trajectories in different phase spaces and highlighting the limitations of canonical transformations in such contexts.

Contribution

It introduces a Hamiltonian for dissipative systems that demonstrates the distinct nature of trajectories in different phase spaces and the failure of canonical transformations to resolve these behaviors.

Findings

Trajectories in (x, p) space exhibit counter-intuitive behavior.

Canonical transformations do not rectify the unusual trajectory behavior.

The Hamiltonian approach clarifies the differences in phase space dynamics for dissipative systems.

Abstract

A Hamiltonian for a one-dimensional (1-D) dissipative system is given which shows that the trajectories in the spaces ($x,\dot x$) and ($x,p$) are completely different. The trajectory in the space ($x,p$) has an unexpected contra-intuitive behavior, and a canonical transformation does not solve this behavior.