On frequencies of small oscillations of some dynamical systems associated with root systems

TL;DR

This paper generalizes a formula for small oscillation frequencies from the Sutherland system to arbitrary root systems, broadening the understanding of dynamical systems linked to root structures.

Contribution

It extends the known oscillation frequency formula from the $A_l$ case to all root systems, providing a more comprehensive theoretical framework.

Findings

Derived a generalized formula for oscillation frequencies

Applied the formula to various root systems

Enhanced understanding of dynamical systems associated with root structures

Abstract

In the paper by F. Calogero and author [Commun. Math. Phys. 59 (1978) 109-116] the formula for frequencies of small oscillations of the Sutherland system ($A_l$ case) was found. In present note the generalization of this formula for the case of arbitrary root system is given.