Geodesic
Systems with dissipation with a finite number of degrees of freedom: analysis and integrability. III. Systems on the tangent bundles of smooth $n$-dimensional manifolds
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 213 (2022), pp. 96-109

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This paper is the third part of a survey on the integrability of systems with a large number $n$ of degrees of freedom (the first part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 211 (2022), pp. 41–74; the second part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 212 (2022), pp. 139–148). The review consists of three parts. In the first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. The second part is devoted to more general dynamical systems on the tangent bundles to the $n$-dimensional sphere. In this third part, we discuss dynamical systems on the tangent bundles to smooth manifolds of a sufficiently wide class. Theorems on sufficient conditions for the integrability of the considered dynamical systems in the class of transcendental functions are proved.
Keywords: dynamical system with a large number of degrees of freedom, integrability, transcendental first integral. 
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     title = {Systems with dissipation with a finite number of degrees of freedom: analysis and integrability.  {III.} {Systems} on the tangent bundles of smooth $n$-dimensional manifolds},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {96--109},
     publisher = {mathdoc},
     volume = {213},
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M. V. Shamolin. Systems with dissipation with a finite number of degrees of freedom: analysis and integrability.  III. Systems on the tangent bundles of smooth $n$-dimensional manifolds. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 213 (2022), pp. 96-109. http://geodesic.mathdoc.fr/item/INTO_2022_213_a9/