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Rigorous lower bounds for the domains of definition of extended generating functions

B. Erdelyi, J. Hoefkens, M. Berz

TL;DR

This work develops a framework to obtain rigorous, lower bounds on the domains of definition for extended generating functions of symplectic maps by combining the extended generating function theory with high-order Taylor-model methods. By linking generator invertibility to the invertibility of specific map components and parameterizing generators with symmetric matrices $S$, the authors derive practical bounds and demonstrate that, in many cases, the domains are large enough to cover dynamic apertures relevant for long-term Hamiltonian simulations. Four diverse examples—from polynomial maps to the Fermi-Pasta-Ulam system and an accelerator cell—illustrate that small $S$ often yields robust, verifiable domains, enabling reliable, verified long-term tracking. The results have direct implications for accelerator physics and Hamiltonian dynamics, providing verifiable guarantees on the applicability of generating-function based tracking methods and enabling accurate enclosure of the true dynamics over practically relevant regions.

Abstract

The recently developed theory of extended generating functions of symplectic maps are combined with methods to prove invertibility via high-order Taylor model methods to obtain rigorous lower bounds for the domains of definition of generating functions. The examples presented suggest that there are considerable differences in the size of the domains of definition of the various types of generating functions studied. Furthermore, the types of generating functions for which, in general, large domains of definitions can be guarantied were identified. The resulting domain sizes of these generators are usually sufficiently large to be useful in practice, for example for the long-term simulations of accelerator and other Hamiltonian system dynamics.

Rigorous lower bounds for the domains of definition of extended generating functions

TL;DR

This work develops a framework to obtain rigorous, lower bounds on the domains of definition for extended generating functions of symplectic maps by combining the extended generating function theory with high-order Taylor-model methods. By linking generator invertibility to the invertibility of specific map components and parameterizing generators with symmetric matrices , the authors derive practical bounds and demonstrate that, in many cases, the domains are large enough to cover dynamic apertures relevant for long-term Hamiltonian simulations. Four diverse examples—from polynomial maps to the Fermi-Pasta-Ulam system and an accelerator cell—illustrate that small often yields robust, verifiable domains, enabling reliable, verified long-term tracking. The results have direct implications for accelerator physics and Hamiltonian dynamics, providing verifiable guarantees on the applicability of generating-function based tracking methods and enabling accurate enclosure of the true dynamics over practically relevant regions.

Abstract

The recently developed theory of extended generating functions of symplectic maps are combined with methods to prove invertibility via high-order Taylor model methods to obtain rigorous lower bounds for the domains of definition of generating functions. The examples presented suggest that there are considerable differences in the size of the domains of definition of the various types of generating functions studied. Furthermore, the types of generating functions for which, in general, large domains of definitions can be guarantied were identified. The resulting domain sizes of these generators are usually sufficiently large to be useful in practice, for example for the long-term simulations of accelerator and other Hamiltonian system dynamics.
Paper Structure (10 sections, 2 theorems, 32 equations, 9 figures, 3 tables)

This paper contains 10 sections, 2 theorems, 32 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The extended generating function theory
  3. Rigorous Numerical Analysis with Taylor Models
  4. Guaranteed Invertibility of Maps
  5. Verified Enclosures of Flows
  6. Examples
  7. A cubic $2D$ symplectic map
  8. A $4D$ symplectic map
  9. The Fermi-Pasta-Ulam system
  10. An accelerator cell

Key Result

Theorem 1

Let ${\cal M}$ be a symplectic map. Then, for every point $z$ there is a neighborhood of $z$ such that ${\cal M}$ can be represented by functions $F$ via the relation where $\alpha =\left( \alpha _{1},\alpha _{2}\right) ^{T}$ is any conformal symplectic map such that Conversely, let $F$ be a twice continuously differentiable function. Then, the map ${\cal M}$ defined by is symplectic. The matri

Figures (9)

  • Figure 1: Tracking picture of the cubic two dimensional symplectic map, and the box of guarantied invertibility of the generator associated with $S=0$.
  • Figure 2: $(q_1,p_1)$ tracking picture of the four dimensional symplectic polynomial, for two initial conditions launched along the $q_1$ axis close to the origin.
  • Figure 3: $(a)$$(q_1,p_1)$, and $(b)$$(q_2,p_2)$ tracking pictures of the four dimensional symplectic polynomial for two particles (launched along the $q_1$ and $q_2$ axes respectively) close to the dynamic aperture, and the box of guarantied invertibility of the generator associated with $S=0$.
  • Figure 4: The Fermi-Pasta-Ulam system used in our example.
  • Figure 5: $(a)$$(q_1,p_1)$, and $(b)$$(q_1,q_2)$ tracking pictures of the half period map of the Fermi-Pasta-Ulam system for a particle launched along the $q_1$ axis. The box of guarantied invertibility of the generator associated with $S=0$ extends to at least $[-1,1]$ in every direction.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1: Existence of extended generating functions
  • Theorem 2: Invertibility from First Derivatives