St\"ackel and Eisenhart lifts, Haantjes geometry and Gravitation
Math · 0.90Rendering · 0.70
Summary · qwen2.5:32b
The article explores the use of generalized Stäckel geometry for constructing new classes of integrable Hamiltonian systems via Stäckel lifts, extending Eisenhart lifts to include both Riemannian and Lorentzian types. It demonstrates that these systems possess a symplectic-Haantjes structure, illustrated through applications in magnetic systems separable in cylindrical coordinates using modified Stäckel bases. The study also suggests potential applications of momentum-dependent lifting matrices in modified gravity theories and Hamilton-Finsler geometries.
Suggested post angle
This article discusses St"ackel and Eisenhart lifts, a topic related to Haantjes geometry and Gravitation. It involves integrable systems, momentum-dependent St"ackel matrices, symplectic-Haantjes structures, and applications in magnetic systems separable in cylindrical coordinates. These concepts are relevant to both rendering (real-time rendering, shader, GPU) and math (geometry, algebra).
Excerpt
arXiv:2509.19950v3 Announce Type: replace
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.