Coordinate-free formulation of the deformed Toda Hamiltonian

Rewrite the classical deformed Toda Hamiltonian in coordinate-free language despite the quadratic dependence on the momenta in its potential term.

Background

The paper introduces a classical long-range deformation of the open Toda Hamiltonian, with interactions extending over all positive roots of the root system A_{n-1}. The deformation weights are related to the lengths of reflections in the symmetric group, which suggests a root-theoretic interpretation analogous to coordinate-free formulations of ordinary Toda systems.

The authors explicitly state that the quadratic momentum dependence in the potential term prevents them from rewriting the Hamiltonian in coordinate-free language. This is an unresolved structural problem concerning the geometric and root-theoretic description of the deformed system.

References

However, we do not know how to rewrite the Hamiltonian Hamp^2 in the coordinate free language due to the quadratic in momenta term in the "potential" part of the Hamiltonian.

Hamp^2:

H=∑i=1npi22−∑i<jS2(j−i−1)(1+Spi)(1−Spj)xixj=H122−H2.H = \sum\limits_{i = 1}^n \frac{p_i^2}{2} - \sum\limits_{i<j}S^{2(j-i-1)}(1+S p_i)(1 - S p_j) \frac{x_i}{x_j} = \frac{H_1^2}{2} - H_2.

— Integrability of the deformed Toda systems  (2609.20679 - Vasilev, 17 Sep 2026) in Remark 1, Section 3.1 (Classical deformed open Toda chain)