Energy self-balance as the physical basis of orbit quantization

TL;DR

This paper demonstrates that energy balance in dissipative systems leads to orbit quantization, deriving conditions similar to quantum rules and applying them to hydrodynamic analogs like walking droplets.

Contribution

It introduces a novel physical basis for orbit quantization based on energy self-balance and phase space area preservation, linking classical dynamics to quantum-like spectra.

Findings

Energy work along stable limit cycles is zero, ensuring energy preservation.

Derived quantization conditions resemble Wilson Sommerfeld rules.

Validated the theory with hydrodynamic quantum analog experiments.

Abstract

We show that work done by the non conservative forces along a stable limit cycle attractor of a dissipative dynamical system is always equal to zero. Thus, mechanical energy is preserved on average along periodic orbits. This balance between energy gain and energy loss along different phases of the self sustained oscillation is responsible for the existence of quantized orbits in such systems. Furthermore, we show that the instantaneous preservation of projected phase space areas along quantized orbits describes the neutral dynamics of the phase, allowing us to derive from this equation the Wilson Sommerfeld like quantization condition. We apply our general results to near Hamiltonian systems, identifying the fixed points of Krylov Bogoliubov radial equation governing the dynamics of the limit cycles with the zeros of the Melnikov function. Moreover, we relate the instantaneous preservation of the phase space area along the quantized orbits to the second Krylov Bogoliubov equation describing the dynamics of the phase. We test the two quantization conditions in the context of hydrodynamic quantum analogs, where a megastable spectra of quantized orbits have recently been discovered. Specifically, we use a generalized pilot wave model for a walking droplet confined in a harmonic potential, and find a countably infinite set of nested limit cycle attractors representing a classical analog of quantized orbits. We compute the energy spectrum and the eigenfunctions of this self excited system.