Geodesic
Application of the Kovacic algorithm to the study of the motion of a heavy rigid body with a fixed point in the Hess case
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 202 (2021), pp. 10-42

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In 1890, W. Hess found a new particular case of the integrable Euler–Poisson equations of the motion of a heavy rigid body with a fixed point. In 1892, P. A. Nekrasov proved that the solution of the problem of the motion of a heavy rigid body with a fixed point under the Hess conditions can be reduced to integrating a second-order linear equation with variable coefficients. In this paper, we derive the corresponding second-order equation and reduce its coefficients to the rational form. Then, using the Kovacic algorithm, we examine the existence of Liouville solutions of the corresponding second-order linear equation. We prove that Liouville solutions can exist only in two cases: in the case corresponding to the Lagrange case of the motion of a rigid body with a fixed point and in the case where the area integral is equal to zero.
Keywords: body with a fixed point, Hess case, Kovacic algorithm.
Mots-clés : Liouville solution 
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     title = {Application of the {Kovacic} algorithm to the study of the motion of a heavy rigid body with a fixed point in the {Hess} case},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {10--42},
     publisher = {mathdoc},
     volume = {202},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2021_202_a1/}
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A. S. Kuleshov. Application of the Kovacic algorithm to the study of the motion of a heavy rigid body with a fixed point in the Hess case. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 202 (2021), pp. 10-42. http://geodesic.mathdoc.fr/item/INTO_2021_202_a1/