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Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation

Ondřej Kubů, Piergiulio Tempesta

TL;DR

The paper develops the Stäckel lift as a unified generalization of the Eisenhart lift that leverages generalized Stäckel matrices to generate new integrable and separable Hamiltonian systems in higher dimensions. It establishes that the Eisenhart lift is a special case of the Stäckel lift and shows that momentum-dependent lifting matrices induce non-projectable Haantjes structures on the lifted space, linking integrability, separability, and Haantjes geometry. The authors provide concrete illustrations, including lifted systems separable in 3D, Smorodinsky–Winternitz examples, and cylindrical magnetic systems, and they demonstrate connections to gravitational waves via Platonic/pp-wave geometries and to magnetic/cylindrical separability in a geometric Haantjes framework. They also extend the construction to iterated geodesic lifts, non-geodesic flows, and gravitational-wave spacetimes, highlighting potential applications to modified gravity theories and Lorentzian geometries with wave-front structure. Overall, the work offers a versatile, geometry-driven toolkit for producing and analyzing integrable, separable, and physically meaningful Hamiltonian systems across Riemannian, Lorentzian, and gauge‑deformed contexts.

Abstract

We study lifts of integrable systems by means of generalized Stäckel geometry. To this aim, we present the notion of Stäckel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The Stäckel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. Moreover, we prove that Hamiltonian systems constructed through momentum-dependent Stäckel matrices are naturally endowed with a non-trivial symplectic-Haantjes structure. We further illustrate applications to magnetic systems separable in cylindrical coordinates; we describe them within the Stäckel framework by means of modified Stäckel basis. We also show that explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries, with potential applications in modified gravity theories.

Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation

TL;DR

The paper develops the Stäckel lift as a unified generalization of the Eisenhart lift that leverages generalized Stäckel matrices to generate new integrable and separable Hamiltonian systems in higher dimensions. It establishes that the Eisenhart lift is a special case of the Stäckel lift and shows that momentum-dependent lifting matrices induce non-projectable Haantjes structures on the lifted space, linking integrability, separability, and Haantjes geometry. The authors provide concrete illustrations, including lifted systems separable in 3D, Smorodinsky–Winternitz examples, and cylindrical magnetic systems, and they demonstrate connections to gravitational waves via Platonic/pp-wave geometries and to magnetic/cylindrical separability in a geometric Haantjes framework. They also extend the construction to iterated geodesic lifts, non-geodesic flows, and gravitational-wave spacetimes, highlighting potential applications to modified gravity theories and Lorentzian geometries with wave-front structure. Overall, the work offers a versatile, geometry-driven toolkit for producing and analyzing integrable, separable, and physically meaningful Hamiltonian systems across Riemannian, Lorentzian, and gauge‑deformed contexts.

Abstract

We study lifts of integrable systems by means of generalized Stäckel geometry. To this aim, we present the notion of Stäckel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The Stäckel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. Moreover, we prove that Hamiltonian systems constructed through momentum-dependent Stäckel matrices are naturally endowed with a non-trivial symplectic-Haantjes structure. We further illustrate applications to magnetic systems separable in cylindrical coordinates; we describe them within the Stäckel framework by means of modified Stäckel basis. We also show that explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries, with potential applications in modified gravity theories.

Paper Structure

This paper contains 26 sections, 5 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the Haantjes geometry
  3. Haantjes algebras
  4. The symplectic-Haantjes manifolds
  5. Haantjes chains
  6. Haantjes coordinates
  7. The Stäckel lift as a generalization of the Eisenhart lift
  8. The Eisenhart lift
  9. Preliminaries on the Stäckel geometry
  10. The Stäckel lift
  11. Integrable Hamiltonian systems from the Stäckel lift
  12. Lifted Systems separable in 3D
  13. The Riemannian Eisenhart lift as a Stäckel lift
  14. The Stäckel lift of Smorodinsky-Winternitz systems
  15. Haantjes geometry and Stäckel lifts
  16. ...and 11 more sections

Key Result

Proposition 1

TT2022AMPA Given a $2n$-dimensional $\omega \mathscr{H}$ manifold $M$, every generalized eigen--distribution $ker(\boldsymbol{K}-l_{i } \boldsymbol{I})^{{r_i}}$, $r_i \in \mathbb{N}$, is of even rank. Therefore, the geometric and algebraic multiplicities of each eigenvalue $l_{i }(\boldsymbol{x})$ a

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Definition 6
  • Proposition 1
  • Definition 7
  • Corollary 1
  • ...and 17 more