Abstract:
We discuss the properties of second-order Killing tensors in three-dimensional Euclidean space that guarantee the existence of a third integral of motion ensuring the Liouville integrability of the corresponding equations of motion. We prove that in addition to the linear Noether and quadratic Stäckel integrals of motion, there are integrable systems with two quadratic integrals of motion and one fourth-order integral of motion in momenta. A generalization to n-dimensional case and to deformations of the standard flat metric is proposed.
Keywords:
Hamilton–Jacobi equations, separation of variables, Killing tensors.
This publication is cited in the following 4 articles:
A. V. Tsiganov, “On rotation invariant integrable systems”, Izv. Math., 88:2 (2024), 389–409
Andrey V. Tsiganov, “Rotations and Integrability”, Regul. Chaotic Dyn., 29:6 (2024), 913–930
A. V. Tsiganov, E. O. Porubov, “On a class of quadratic conservation laws for Newton equations in Euclidean space”, Theoret. and Math. Phys., 216:2 (2023), 1209–1237
E. O. Porubov, A. V. Tsiganov, “Second order Killing tensors related to symmetric spaces”, Journal of Geometry and Physics, 191 (2023), 104911