Second order Killing tensors related to symmetric spaces

TL;DR

This paper explores second-order Killing tensors in symmetric spaces, focusing on their role in integrability of Hamiltonian systems with polynomial integrals of motion beyond quadratic order.

Contribution

It introduces the analysis of second-order Killing tensors linked to quadratic integrals and their relation to symmetries involving rotations and translations in Euclidean space.

Findings

Identification of second-order Killing tensors related to polynomial integrals

Connection between these tensors and symmetries in Euclidean space

Extension of integrability conditions beyond quadratic integrals

Abstract

We discuss the pairs of quadratic integrals of motion belonging to the $n$-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisenhart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta. The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.