Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University550010.18287/2541-7525-2017-23-3-41-64UnknownON A PENDULUM MOTION IN MULTI-DIMENSIONAL SPACE. PART 1. DYNAMICAL SYSTEMSShamolinM. V.morenov.sv@ssau.ruInstitute of Mechanics, Lomonosov Moscow State University1511201723341641501201815012018Copyright © 2017, Shamolin M.V.2017<p>In the proposed cycle of work, we study the equations of the motion of dynamically symmetric fixed n-dimensional rigid bodiespendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free n-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In thit work, we derive the general multi-dimensional dynamic equations of the systems under study.</p>многомерное твердое тело, неконсервативное поле сил, динамическая система, случаи интегрируемости.multi-dimensional rigid body, non-conservative force field, dynamical system, case of integrability.[[1] Shamolin M.V. Sluchai integriruemosti, sootvetstvuiushchie dvizheniiu maiatnika na ploskosti [Cases of integrability corresponding to the pendulum motion on the plane]. Vestnik SamGU. Estestvennonauchnaia seriia [Vestnik of Samara State University. Natural Science Series], 2015, no. 10(132), pp. 91–113 [in Russian].][[2] Shamolin M.V. Sluchai integriruemosti, sootvetstvuiushchie dvizheniiu maiatnika v trekhmernom prostranstve [Cases of integrability corresponding to the pendulum motion on the three-dimensional space]. Vestnik SamGU. Estestvennonauchnaia seriia [Vestnik of Samara State University. Natural Science Series], 2016, no. 3–4, pp. 75–97 [in Russian].][[3] Shamolin M.V. Mnogoobrazie sluchaev integriruemosti v dinamike malomernogo i mnogomernogo tverdogo tela v nekonservativnom pole [Variety of cases of integrability in dynamics of lower-, and multi-dimensional body in nonconservative field]. Itogi nauki i tekhniki. Ser.: ”Sovremennaia matematika i ee prilozheniia. Tematicheskie obzory”. T. 125. ”Dinamicheskie sistemy”. [Journal of Mathematical Sciences. Vol 125. Dynamical Systems], 2013, pp. 5–254 [in Russina].][[4] Pokhodnya N.V., Shamolin M.V. Nekotorye usloviia integriruemosti dinamicheskikh sistem v transtsendentnykh funktsiiakh [Some cases of integrability of dynamic systems in transcedent functions]. Vestnik SamGU. Estestvennonauchnaia seriia [Vestnik of Samara State University. Natural Science Series], 2013, no. 9/1(110), pp. 35–41 [in Russian].][[5] Shamolin M.V. Mnogoobrazie tipov fazovykh portretov v dinamike tverdogo tela, vzaimodeistvuiushchego s soprotivliaiushcheisia sredoi [Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium]. Doklady RAN [Physics Doklady], 1996, Vol. 349, no. 2, pp. 193–197 [in Russian].][[6] Shamolin M.V. Dinamicheskie sistemy s peremennoi dissipatsiei: podkhody, metody, prilozheniia [Dynamical Systems With Variable Dissipation: Approaches, Methods, and Applications]. Fund. i prikl. mat. [Journal of Mathematical Sciences], 2008, Vol. 14, no. 3, pp. 3–237 [in Russian].][[7] Arnold V.I., Kozlov V.V., Neyshtadt A.I. Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki [Mathematical aspects in classical and celestial mechanics]. M.: VINITI, 1985, 304 p. [in Russian].][[8] Trofimov V.V. Simplekticheskie struktury na gruppakh avtomorfizmov simmetricheskikh prostranstv [Symplectic structures on symmetruc spaces automorphysm groups]. Vestn. Mosk. un–ta. Ser. 1. Matematika. Mekhanika [Moscow University Mathematics Bulletin], 1984, no. 6, pp. 31–33 [in Russian].][[9] Trofimov V.V., Shamolin M.V. Geometricheskie i dinamicheskie invarianty integriruemykh gamil’tonovykh idissipativnykh sistem [Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems].][Fund. i prikl. mat. [Journal of Mathematical Sciences], 2010, Vol. 16, no. 4, pp. 3–229 [in Russian].][[10] Shamolin M.V. Metody analiza dinamicheskikh sistem s peremennoi dissipatsiei v dinamike tverdogo tela [Methods of analysis of various dissipation dynamical systems in dynamics of a rigid body]. M.: Izd-vo ”Ekzamen ”, 2007, 352 p. [in Russian].][[11] Shamolin M.V. Nekotorye model’nye zadachi dinamiki tverdogo tela pri vzaimodeistvii ego so sredoi [Some model problems of dynamics for a rigid body interacting with a medium]. Prikl. mekhanika [International Applied Mechanics], 2007, Vol. 43, no. 10, pp. 49–67 [in Russian].][[12] Shamolin M.V. Novye sluchai integriruemosti sistem s dissipatsiei na kasatel’nykh rassloeniiakh k dvumernoi I trekhmernoi sferam [New Cases of Integrable Systems with Dissipation on Tangent Bundles of Two- and Three-Dimensional Spheres]. Doklady RAN [Physics Doklady], 2016, Vol. 471, no. 5, pp. 547–551 [in Russian].][[13] Shamolin M.V. Novye sluchai integriruemykh sistem s dissipatsiei na kasatel’nom rassloenii k mnogomernoi sfere [New Cases of Integrable Systems with Dissipation on a Tangent Bundle of a Multidimensional Sphere]. Doklady RAN [Physics Doklady], 2017, Vol. 474, no. 2, pp. 177–181 [in Russian].][[14] Shamolin M.V. Novye sluchai integriruemykh sistem s dissipatsiei na kasatel’nom rassloenii dvumernogo mnogoobraziia [New Cases of Integrable Systems with Dissipation on a Tangent Bundle of a Two-Dimensional Manifold]. Doklady RAN [Physics Doklady], 2017, Vol. 475, no. 5, pp. 519–523 [in Russian].]