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Foundations of ghost stability

Verónica Errasti Díez, Jordi Gaset Rifà, Georgina Staudt

TL;DR

Foundations of ghost stability addresses the problem of ensuring global stability for higher-order (ghost) dynamical systems whose Hamiltonians are not bounded from below. It introduces a unifying framework based on conserved quantities, extending energy- and coercivity-based methods, and proves a novel confining-quantity criterion that guarantees $G1$ stability in the compact sense. The work connects stability to symmetry structures and integrability (Liouville-Arnold), and provides explicit ghost-model examples that broaden the known stable families. It highlights potential implications for field theory and quantization, offering analytic tools that go beyond traditional Ostrogradsky arguments and opening avenues for inter-disciplinary dynamics and integrability research.

Abstract

We present a new method to analytically prove global stability in ghost-ridden dynamical systems. Our proposal encompasses all prior results and consequentially extends them. In particular, we show that stability can follow from a conserved quantity that is unbounded from below, contrary to expectation. Novel examples illustrate all our results. Our findings take root on a careful examination of the literature, here comprehensively reviewed for the first time. This work lays the mathematical basis for ulterior extensions to field theory and quantization, and it constitutes a gateway for inter-disciplinary research in dynamics and integrability.

Foundations of ghost stability

TL;DR

Foundations of ghost stability addresses the problem of ensuring global stability for higher-order (ghost) dynamical systems whose Hamiltonians are not bounded from below. It introduces a unifying framework based on conserved quantities, extending energy- and coercivity-based methods, and proves a novel confining-quantity criterion that guarantees stability in the compact sense. The work connects stability to symmetry structures and integrability (Liouville-Arnold), and provides explicit ghost-model examples that broaden the known stable families. It highlights potential implications for field theory and quantization, offering analytic tools that go beyond traditional Ostrogradsky arguments and opening avenues for inter-disciplinary dynamics and integrability research.

Abstract

We present a new method to analytically prove global stability in ghost-ridden dynamical systems. Our proposal encompasses all prior results and consequentially extends them. In particular, we show that stability can follow from a conserved quantity that is unbounded from below, contrary to expectation. Novel examples illustrate all our results. Our findings take root on a careful examination of the literature, here comprehensively reviewed for the first time. This work lays the mathematical basis for ulterior extensions to field theory and quantization, and it constitutes a gateway for inter-disciplinary research in dynamics and integrability.
Paper Structure (30 sections, 11 theorems, 61 equations, 2 figures)

This paper contains 30 sections, 11 theorems, 61 equations, 2 figures.

Table of Contents

  1. Motivation
  2. Organization.
  3. State of the art: multifacetedness of ghost stability
  4. Global notions of stability.
  5. Local notions of stability.
  6. Main classical results
  7. Comments on quantization
  8. Definitions: the big picture of global stability
  9. Proving G1: the pivotal role of conserved quantities
  10. When does energy work?
  11. The case of ghost systems
  12. Main result: generalization of the method
  13. Relation to symmetries
  14. Relation to G2 and Liouville(-Mineur)-Arnold theorem.
  15. Examples
  16. ...and 15 more sections

Key Result

Theorem 1

Given a system $\emph{Sol}$ with $\mathcal{P}=\mathbb{R}^n$, if the energy $E:\mathbb{R}^n\rightarrow \mathbb{R}$ is a twice differentiable conserved quantity that has a minimum and whose Hessian is positive definite everywhere, then the system is G1.

Figures (2)

  • Figure 1: Visualization of the scope of global stability definitions \ref{['def:G1bounded']}, \ref{['def:G1compact']} and \ref{['def:G2formal']}, together with relations between them. These formalize the preexisting notions \ref{['def:G1']} and \ref{['def:G2']}.
  • Figure 2: Archetypal instances of the rich phenomenology of the right-most example in (\ref{['eq:exth3no2']}). The function (\ref{['eq:plottedQ']}) is shown, with its variable $\zeta$ labelling the abscissa. The free parameters have been set to $\omega^2=1$, $\alpha=0.3$, $\nu=7$ and $b=0$ (plot A), $b=1/8$ (plot B), $b=b_c\approx0.55$ (plot C), $b=2$ (plot D). Thereby, the itemization below (\ref{['eq:exth3no2']}) is illustrated. All these are exemplary potentials of conserved quantities that can be used to analytically prove G1 stability in the compact sense of definition \ref{['def:G1compact']}. The lower half portrays radically new proposals, including totally unbounded scenarios.

Theorems & Definitions (28)

  • Definition 1.a
  • Definition 1.b
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Theorem 2
  • Definition 5
  • Theorem 3
  • proof
  • ...and 18 more