abstract 64

Bulletin of Computational Applied Mathematics (Bull CompAMa)

64

A note on the integrability of exceptional potentials via polynomial bi-homogeneous potentials

Primitivo B. Acosta-Humánez, Martha Álvarez-Ramírez, Teresinha J. Stuchi

This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=\alpha (q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with $\alpha\in\mathbb{C}$ and $l=0,1,\dots, k$, called exceptional potentials. Hietarinta \cite{Hietarinta1983} proved that the potentials with $l=0,1,k-1,k$ and $l=k/2$ for $k$ even are polynomial integrable. We present an elementary proof of this fact in the context of the polynomial bi-homogeneous potentials, as was introduced by Combot et al. \cite{Combot2020}. In addition, we take advantage of the fact that we can exchange the exponents to derive an additional first integral for $V_{7,5}$, unknown so far. The paper concludes with a Galoisian analysis for $l=k/2$.

Keywords: Hamiltonian system with two degrees of freedom; Homogeneous potentials; Integrability.

Cite this paper:

Acosta-Humánez P.B., Álvarez-Ramírez M., Stuchi T.J.

A note on the integrability of exceptional potentials via polynomial bi-homogeneous potentials

Bull. Comput. Appl. Math. (Bull CompAMa)

Vol. 9, No.2, pp.59-75, 2021.