[Paper Review] Ternary Poisson algebra for the non degenerate three dimensional Kepler Coulomb potential

Y. Tanoudis, C. Daskaloyannis|ArXiv.org|Feb 2, 2009

Quantum Mechanics and Non-Hermitian Physics12 references5 citations

TL;DR

This paper establishes that the non-degenerate three-dimensional Kepler-Coulomb potential, a superintegrable system with quadratic and quartic integrals of motion, closes under a ternary parafermionic-like Poisson algebra. Using the Kalnins-Kress-Miller '5 to 6' theorem, it proves the existence of a sixth independent quadratic integral, leading to a closed quadratic ternary Poisson algebra structure with five generators, generalizing the algebraic framework of non-degenerate superintegrable systems.

ABSTRACT

In the three dimensional flat space any classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller have proved that, in the case of non degenerate potentials, i.e potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. The Kepler Coulomb potential that was introduced by Verrier and Evans is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the non degenerate case of systems with quadratic integrals.

Motivation & Objective

• To determine the Poisson algebra structure of integrals of motion for the non-degenerate three-dimensional Kepler-Coulomb potential.
• To investigate whether the system's six integrals (five quadratic and one quartic) close under Poisson brackets in a non-trivial algebraic structure.
• To extend the known '5 to 6' theorem for non-degenerate superintegrable systems to the case of the generalized Kepler-Coulomb potential with additional inverse-square terms.
• To classify the resulting algebra as a ternary parafermionic-like Poisson algebra, analogous to other non-degenerate systems with quadratic integrals.

Proposed method

• Application of the Kalnins-Kress-Miller '5 to 6' theorem to prove the existence of a sixth linearly independent quadratic integral in the non-degenerate 3D Kepler-Coulomb system.
• Computation of nested Poisson brackets { {I_i, I_j}, I_k } for all combinations of the five quadratic integrals A1, A2, B1, B2, H and the sixth integral F.
• Derivation of explicit algebraic relations showing that the ternary Poisson bracket { {x_i, x_j}, x_k } closes into linear and quadratic combinations of the generators, confirming the algebraic closure.
• Identification of the resulting algebra as a quadratic ternary Poisson algebra with structure constants matching the parafermionic-like pattern observed in other non-degenerate systems.
• Use of the Hamiltonian H and the five quadratic integrals A1, A2, B1, B2, and F as generators to define the algebraic closure relations.
• Verification of the closure relations through symbolic computation of Poisson brackets involving the full set of integrals, including the quartic integral F.

Experimental results

Research questions

• RQ1 Does the non-degenerate three-dimensional Kepler-Coulomb potential, with inverse-square terms, close under a non-trivial Poisson algebra beyond the standard quadratic algebra?
• RQ2 Can the '5 to 6' theorem be applied to systems with higher-order integrals, such as the generalized Kepler-Coulomb potential with quartic integrals?
• RQ3 Is the Poisson algebra of the integrals in this system isomorphic to a known algebraic structure, such as a ternary parafermionic-like algebra?
• RQ4 How do the nested Poisson brackets { {I_i, I_j}, I_k } behave in this system, and do they close into linear and quadratic combinations of the generators?
• RQ5 What is the algebraic structure of the Poisson algebra when the system has both quadratic and quartic integrals of motion?

Key findings

• The non-degenerate 3D Kepler-Coulomb potential possesses a sixth quadratic integral of motion, linearly independent from the five existing quadratic integrals, confirming the '5 to 6' theorem.
• The Poisson algebra of the six integrals (five quadratic and one quartic) closes under nested Poisson brackets, forming a closed quadratic ternary Poisson algebra.
• The resulting algebra is isomorphic to a ternary parafermionic-like Poisson algebra, characterized by specific structure constants in the ternary bracket { {x_i, x_j}, x_k }.
• The algebra exhibits a 'Π' structure with three subalgebras, each corresponding to a classical superintegrable system with two Hamiltonians.
• The Poisson brackets of the generators yield explicit expressions involving the Hamiltonian H, the integrals A1, A2, B1, B2, and F, confirming closure into linear and quadratic combinations.
• The system's algebraic structure suggests that degenerate 3D potentials may be related to non-degenerate systems with higher-order integrals, though this remains an open direction.