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arXiv · 2509.19950

St\"ackel and Eisenhart lifts, Haantjes geometry and Gravitation

Ondřej Kubů·Piergiulio Tempesta
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Abstract

We study lifts of integrable systems by means of generalized St\"ackel geometry. To this end, we present the notion of St\"ackel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The St\"ackel 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 St\"ackel 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 St\"ackel framework by means of modified St\"ackel bases. Finally, we show that explicitly momentum-dependent lifting matrices generate Platonic-wave geometries with potential applications in modified gravity theories, or momentum-dependent metrics of Hamilton and Finsler geometries.

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Ondřej Kubů, Piergiulio Tempesta. 2025-09-24. St\"ackel and Eisenhart lifts, Haantjes geometry and Gravitation. https://doi.org/10.1098/rspa.2025.1103

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