Integrability of potentials of degree k \≠ \± 2. Second order\n variational equations between Kolchin solvability and Abelianity

2013/03/23 by Guillaume Duval, Duval, Guillaume, Andrzej J. Maciejewski +1
Mathematics · Physics and Astronomy · #34M35 #37J30 #37J35 #70H07 #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1303.5829

openalex publication_date 2013/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In our previous paper: Integrability of Homogeneous potentials of degree k =\n\± 2. An application of higher order variational equations, we tried to\nextract some particular structures of the higher variational equations (the\nVE_p for p >1 ), along particular solutions of some Hamiltonian systems.\nThen, we use them to get new Galois obstructions to the integrability of\nnatural Hamiltonian with potential of degree k = \± 2. In the present work,\nwe apply the results of the previous paper, to the complementary cases, when\nthe degrees of the potentials are relative integers k, with |k| >2. Since\nthese cases are much more general and complicated, we reduce our study only to\nthe second variational equation VE₂.\n

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