On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

Case	Potential of degree 4
V_i	α(q₂ − iq₁)ⁱ(q₂+iq₁)⁴⁻ⁱ for i = 0,1,2,3,4.
V₅	$\begin{matrix} α q_{2}^{4} \end{matrix}$ $\begin{array}{} \alpha q_2^4 \end{array} $
V₆	$\begin{matrix} α q_{1}^{4} / 4 + q_{2}^{4} \end{matrix}$ $\begin{array}{} \alpha q_1^4/4+q_2^4 \end{array} $
V₇	$\begin{matrix} 4 q_{1}^{4} + 3 q_{1}^{2} q_{2}^{2} + q_{2}^{4} / 4 \end{matrix}$ $\begin{array}{} 4q_1^4+3q_1^2q_2^2+q_2^4/4 \end{array} $
V₈	$\begin{matrix} 2 q_{1}^{4} + 3 q_{1}^{2} q_{2}^{2} / 2 + q_{2}^{4} / 4 \end{matrix}$ $\begin{array}{} 2q_1^4 + 3q_1^2 q_2^2/2+q_2^4/4 \end{array} $
V₉	$\begin{matrix} (q_{1}^{2} + q_{2}^{2})^{2} / 4 \end{matrix}$ $\begin{array}{} (q_1^2+ q_2^2)^2/4 \end{array} $
V₁₀	$\begin{matrix} - q_{1}^{2} (q_{1} + i q_{2})^{2} + (q_{1}^{2} + q_{2}^{2})^{2} / 4 \end{matrix}$ $\begin{array}{} -q_1^2(q_1+ i q_2)^2+(q_1^2+ q_2^2)^2/4 \end{array} $

Non–equivalent integrable homogeneous potentials of degree 3_ Here i=−1_$\begin{array}{} i= \sqrt{-1}_ \end{array} $

Case	Potentials of degree 3
V₁	$\begin{matrix} q_{1}^{3} \end{matrix}$ $\begin{array}{} q_1^3 \end{array} $
V₂	$\begin{matrix} q_{1}^{3} / 3 + c q_{2}^{3} / 3 \end{matrix}$ $\begin{array}{} q_1^3/3+cq_2^3/3 \end{array} $
V₃	$\begin{matrix} a q_{1}^{3} / 3 + q_{1}^{2} q_{2} / 2 + q_{2}^{3} / 6 \end{matrix}$ $\begin{array}{} aq_1^3/3+q_1^2q_2/2+q_2^3/6 \end{array} $
V₄	$\begin{matrix} q_{1}^{2} q_{2} / 2 + q_{2}^{3} \end{matrix}$ $\begin{array}{} q_1^2q_2/2 +q_2^3 \end{array} $
V₅	$\begin{matrix} \pm 7 q_{1}^{3} i / 15 + q_{1}^{2} q_{2} / 2 + q_{2}^{3} / 15 \end{matrix}$ $\begin{array}{} \pm 7q_1^3 i/15+q_1^2q_2/2+q_2^3/15 \end{array} $
V₆	$\begin{matrix} q_{1}^{2} q_{2} / 2 + 8 q_{2}^{3} / 3 \end{matrix}$ $\begin{array}{} q_1^2q_2/2 +8q_2^3/3 \end{array} $
V₇	$\begin{matrix} \pm 17 \sqrt{14} q_{1}^{3} i / 90 + q_{1}^{2} q_{2} / 2 + q_{2}^{3} / 45 \end{matrix}$ $\begin{array}{} \pm 17 \sqrt{14}q_1^3 i/90+q_1^2q_2/2+q_2^3/45 \end{array} $
V₈	$\begin{matrix} \pm \sqrt{3} q_{1}^{3} i / 18 + q_{1}^{2} q_{2} / 2 + q_{2}^{3} \end{matrix}$ $\begin{array}{} \pm \sqrt{3}q_1^3 i/18+q_1^2q_2/2+q_2^3 \end{array} $
V₉	$\begin{matrix} \pm 3 \sqrt{3} q_{1}^{3} i / 10 + q_{1}^{2} q_{2} / 2 + q_{2}^{3} / 45 \end{matrix}$ $\begin{array}{} \pm 3 \sqrt{3}q_1^3 i/10+q_1^2q_2/2+q_2^3/45 \end{array} $
V₁₀	$\begin{matrix} \pm 11 \sqrt{3} q_{1}^{3} i / 45 + q_{1}^{2} q_{2} / 2 + q_{2}^{3} / 10 \end{matrix}$ $\begin{array}{} \pm 11\sqrt{3}q_1^3 i/45+q_1^2q_2/2+q_2^3/10 \end{array} $

All non–equivalent integrable Hamiltonian systems (1) having homogeneous polynomial potentials of degree 3 with their polynomial first integrals independent of the Hamiltonian_

Potential	First integral
V₀	F₀ = p₁ − p₂i,
V₁	$\begin{matrix} F_{1} = 9 p_{1}^{2} i + 6 p_{1} p_{2} + 3 p_{2}^{2} i - 16 a q_{1}^{3} + 24 a q_{1}^{2} q_{2} i + 8 a q_{2}^{3} i, \end{matrix}$ $\begin{array}{} F_1= 9 p_1^2 i + 6 p_1 p_2 + 3 p_2^2 i - 16 a q_1^3 + 24 a q_1^2 q_2 i + 8 a q_2^3 i, \end{array} $
V₂	$\begin{matrix} F_{2} = - 9 p_{1}^{2} i + 6 p_{1} p_{2} - 3 p_{2}^{2} i - 16 a q_{1}^{3} - 24 a q_{1}^{2} q_{2} i - 8 a q_{2}^{3} i, \end{matrix}$ $\begin{array}{} F_2=-9 p_1^2 i+ 6 p_1 p_2 - 3p_2^2 i- 16 a q_1^3 - 24 a q_1^2 q_2 i - 8 a q_2^3 i, \end{array} $
V₃	F₃ = p₁ + p₂i,
V₄	F₄ = p₂,
V₅	$\begin{matrix} F_{5} = 3 p_{1}^{2} + 2 q_{1}^{3}, \end{matrix}$ $\begin{array}{} F_5=3 p_1^2 + 2 q_1^3, \end{array} $
V₆	$\begin{matrix} F_{6} = - 8 p_{1} p_{2} q_{1} + 8 p_{1}^{2} q_{2} - q_{1}^{2} (q_{1}^{2} + 4 q_{2}^{2}), \end{matrix}$ $\begin{array}{} F_6=-8 p_1 p_2 q_1 + 8 p_1^2 q_2 - q_1^2 (q_1^2 + 4 q_2^2), \end{array} $
V₇	$\begin{matrix} F_{7} = 72 p_{1}^{4} - 36 p_{1} p_{2} q_{1}^{3} - 3 q_{1}^{6} + 2 (3 p_{2}^{2} + 16 q_{2}^{3}) (3 p_{2}^{2} + 6 q_{1}^{2} q_{2} + 16 q_{2}^{3}) + 12 p_{1}^{2} (3 p_{2}^{2} + 12 q_{1}^{2} q_{2} + 16 q_{2}^{3}), \end{matrix}$ $\begin{array}{} F_7=72 p_1^4 - 36 p_1 p_2 q_1^3 - 3 q_1^6 + 2 (3 p_2^2 + 16 q_2^3)(3 p_2^2 + 6 q_1^2 q_2 + 16 q_2^3)+ 12 p_1^2 (3 p_2^2 + 12 q_1^2 q_2 + 16 q_2^3), \end{array} $
$\begin{matrix} V_{8}^{\pm} \end{matrix}$ $\begin{array}{} V_8^{\pm} \end{array} $	$\begin{matrix} F_{8}^{\pm} = - q_{1}^{6} \pm 6 \sqrt{3} q_{1}^{5} q_{2} i + 27 q_{1}^{4} q_{2}^{2} \pm 6 \sqrt{3} q_{1}^{3} (p_{1}^{2} + p_{2}^{2} + 2 q_{2}^{3}) i + 54 q_{1}^{2} q_{2} (p_{1}^{2} + p_{2}^{2} + 2 q_{2}^{3}) + 27 (p_{1}^{2} + p_{2}^{2} + 2 q_{2}^{3})^{2} . \end{matrix}$ $\begin{array}{} F_8^{\pm}= -q_1^6 \pm 6 \sqrt{3}q_1^5 q_2 i+ 27 q_1^4 q_2^2 \pm 6 \sqrt{3}q_1^3 (p_1^2 + p_2^2 + 2 q_2^3) i + 54 q_1^2 q_2 (p_1^2 + p_2^2 + 2 q_2^3) + 27 (p_1^2 + p_2^2 + 2 q_2^3)^2. \end{array} $

All non–equivalent integrable Hamiltonian systems (1) having homogeneous polynomial potentials of degree 4 with their polynomial first integral independent of the Hamiltonian_

Potential	First integral
V₀	F₀ = p₁ − p₂i,
V₁	$\begin{matrix} F_{1} = 2 p_{1} p_{2} - 2 p_{2}^{2} i + α q_{1}^{4} i + 6 α q_{1}^{2} q_{2}^{2} i + 8 α q_{1} q_{2}^{3} - 3 α q_{2}^{4} i, \end{matrix}$ $\begin{array}{} F_1= 2 p_1 p_2-2p_2^2 i+\alpha q_1^4 i+6 \alpha q_1^2 q_2^2 i+8 \alpha q_1 q_2^3-3 \alpha q_2^4 i, \end{array} $
V₂	F₂ = p₂q₁ − p₁q₂,
V₃	$\begin{matrix} F_{3} = 2 p_{1} p_{2} + 2 p_{2}^{2} i - α q_{1}^{4} i - 6 α q_{1}^{2} q_{2}^{2} i + 8 α q_{1} q_{2}^{3} + 3 α q_{2}^{4} i, \end{matrix}$ $\begin{array}{} F_3= 2 p_1 p_2+2p_2^2 i-\alpha q_1^4 i-6\alpha q_1^2 q_2^2 i+8 \alpha q_1 q_2^3+3 \alpha q_2^4 i, \end{array} $
V₄	F₄ = p₁ + p₂i,
V₅	$\begin{matrix} F_{5} = p_{2}^{2} + 2 α q_{2}^{4}, \end{matrix}$ $\begin{array}{} F_5= p_2^2+2 \alpha q_2^4, \end{array} $
V₆	$\begin{matrix} F_{6} = p_{2}^{2} + 2 q_{2}^{4}, \end{matrix}$ $\begin{array}{} F_6= p_2^2+2 q_2^4, \end{array} $
V₇	$\begin{matrix} F_{7} = - p_{1} p_{2} q_{2} + p_{2}^{2} q_{1} - 2 q_{1}^{3} q_{2}^{2} - q_{1} q_{2}^{4}, \end{matrix}$ $\begin{array}{} F_7= -p_1 p_2 q_2+p_2^2 q_1-2 q_1^3 q_2^2-q_1 q_2^4, \end{array} $
V₈	$\begin{matrix} F_{8} = p_{1}^{4} + 2 p_{1}^{2} p_{2}^{2} + 8 p_{1}^{2} q_{1}^{4} + 6 p_{1}^{2} q_{1}^{2} q_{2}^{2} + 4 p_{1} p_{2} q_{1} q_{2}^{3} + 8 p_{2}^{2} q_{1}^{4} + 16 q_{1}^{8} + 24 q_{1}^{6} q_{2}^{2} + 12 q_{1}^{4} q_{2}^{4} + 2 q_{1}^{2} q_{2}^{6}, \end{matrix}$ $\begin{array}{} F_8= p_1^4+2 p_1^2 p_2^2+8 p_1^2 q_1^4+6 p_1^2 q_1^2 q_2^2+4 p_1 p_2 q_1 q_2^3+8 p_2^2 q_1^4 +16 q_1^8+24 q_1^6 q_2^2+12 q_1^4 q_2^4+2 q_1^2 q_2^6, \end{array} $
V₉	F₉ = p₁q₂ − p₂q₁,
V₁₀	$\begin{matrix} F_{10} = p_{1}^{2} + 3 p_{1} p_{2} i - 2 p_{2}^{2} + (q_{1} - q_{2} i)^{3} q_{2} . \end{matrix}$ $\begin{array}{} F_{10}= p_1^2 +3 p_1 p_2 i-2p_2^2 +(q_1 -q_ 2 i)^3 q_2. \end{array} $

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On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

Jaume Llibre

Xiang Zhang

Published Online: Dec 31, 2018

Page range: 527 - 536

Received: Sep 24, 2018

Accepted: Dec 24, 2018

DOI: https://doi.org/10.2478/AMNS.2018.2.00041

KeywordsHamiltonian system with 2–degrees of freedom, homogeneous polynomial potential, integrability

© 2018 J. Llibre and X. Zhang, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Non–equivalent integrable homogeneous potentials of degree 4_

Non–equivalent integrable homogeneous potentials of degree 3_ Here i=−1_$\begin{array}{} i= \sqrt{-1}_ \end{array} $

All non–equivalent integrable Hamiltonian systems (1) having homogeneous polynomial potentials of degree 3 with their polynomial first integrals independent of the Hamiltonian_

All non–equivalent integrable Hamiltonian systems (1) having homogeneous polynomial potentials of degree 4 with their polynomial first integral independent of the Hamiltonian_

Keywords
Hamiltonian system with 2–degrees of freedom, homogeneous polynomial potential, integrability