# American Institute of Mathematical Sciences

• Previous Article
Stochastic geodesics and forward-backward stochastic differential equations on Lie groups
• PROC Home
• This Issue
• Next Article
Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions
2013, 2013(special): 105-113. doi: 10.3934/proc.2013.2013.105

## New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies

 1 Department of Mathematics, Engineering, and Computer Science, City University of New York, LaGCC, Long Island City, NY 11101, United States

Received  September 2012 Published  November 2013

In this paper we construct a new class of nonstationary exact solutions for the equations of motion of a classical model of multibody dynamics -- a chain of $n$ heavy rigid bodies that are sequentially coupled by ideal spherical hinges. We establish sufficient conditions for the existence of the solutions and show how the equations of motion can be reduced to quadratures in the case when these conditions are fulfilled.
Citation: Dmitriy Chebanov. New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies. Conference Publications, 2013, 2013 (special) : 105-113. doi: 10.3934/proc.2013.2013.105
##### References:
 [1] D. Bobylev, On a certain particular solution of the differential equations of rotation of a heavy rigid body about a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Estestvozn., 8 (1896), 21-25. [2] A.V. Borisov and I.S. Mamaev, "Dynamics of a rigid body. Hamiltonian methods, integrability, chaos," Institute of Computer Science, Moscow - Izhevsk, 2005. [3] D.A. Chebanov, On a generalization of the problem of similar motions of a system of Lagrange gyroscopes, Mekh. Tverd. Tela, 27 (1995), 57-63. [4] D.A. Chebanov, New dynamical properties of a system of Lagrange gyroscopes, Proceedings of the Institute of Applied Mathematics and Mechanics, 5 (2000), 172-182. [5] D. Chebanov, Exact solutions for motion equations of symmetric gyros system, Multibody Syst. Dyn., 6 (2001), 39-57. [6] D.A. Chebanov, A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the systems's skeleton, Mekh. Tverd. Tela, 41 (2011), 244-254. [7] L. Euler, Du mouvement de rotation des corps solides autour d'un axe variable, Memoires de l'Academie des Sciences de Berlin, XIV (1765), 154-193. [8] G.V. Gorr, Precessional motions in rigid body dynamics and the dynamics of systems of coupled rigid bodies, J. Appl. Math. Mech., 67 (2003), 511-523. [9] G.V. Gorr, L.V. Kudryashova, and L.A. Stepanova, "Classical problems in the theory of solid bodies. Their development and current state," Naukova dumka, Kiev, 1978. [10] G.V. Gorr and V.N. Rubanovskii, A new class of motions of a system of heavy freely connected rigid bodies, J. Appl. Math. Mech., 52 (1988), 551-555. [11] P.V. Kharlamov, The equations of motion of a system of rigid bodies, Mekh. Tverd. Tela, 4 (1972), 52-73. [12] P.V. Kharlamov, Some classes of exact solutions of the problem of the motion of a system of Lagrange gyroscopes, Mat. Fiz., 32 (1982), 63-76. [13] E.I. Kharlamova, A survey of exact solutions of problems of the motion of systems of coupled rigid bodies, Mekh. Tverd. Tela, 26 (1998), 125-138. [14] S.V. Kovalevskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. [15] J.L. Lagrange, "Mechanique Analitique," Veuve Desaint, Paris, 1815. [16] E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about Fixed Point," Springer-Verlag, Berlin, 1965. [17] L. Lilov and N. Vasileva, Steady motion of a system of Lagrange gyroscopes with a tree structure, Teoret. Prilozhna Mekh., XV (1984), 24-34. [18] A.Ya. Savchenko and M.E. Lesina, Particular solution for motion equations of Lagrange gyroscopes system, Mekh. Tverd. Tela, 5 (1973), 27-30. [19] N. Sreenath, Y.G. Oh, P.S. Krishnaprasad, and J.E. Marsden, The dynamics of coupled planar rigid bodies. Part I: Reduction, equilibria and stability, Dynam. Stability Systems, 3 (1988), 25-49. [20] V.A. Steklov, A certain case of motion of a heavy rigid body having a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Lyubit. Estestvozn., 8 (1896), 19-21. [21] J. Wittenburg, "Dynamics of Multibody Systems," Springer-Verlag, Berlin, 2008.

show all references

##### References:
 [1] D. Bobylev, On a certain particular solution of the differential equations of rotation of a heavy rigid body about a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Estestvozn., 8 (1896), 21-25. [2] A.V. Borisov and I.S. Mamaev, "Dynamics of a rigid body. Hamiltonian methods, integrability, chaos," Institute of Computer Science, Moscow - Izhevsk, 2005. [3] D.A. Chebanov, On a generalization of the problem of similar motions of a system of Lagrange gyroscopes, Mekh. Tverd. Tela, 27 (1995), 57-63. [4] D.A. Chebanov, New dynamical properties of a system of Lagrange gyroscopes, Proceedings of the Institute of Applied Mathematics and Mechanics, 5 (2000), 172-182. [5] D. Chebanov, Exact solutions for motion equations of symmetric gyros system, Multibody Syst. Dyn., 6 (2001), 39-57. [6] D.A. Chebanov, A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the systems's skeleton, Mekh. Tverd. Tela, 41 (2011), 244-254. [7] L. Euler, Du mouvement de rotation des corps solides autour d'un axe variable, Memoires de l'Academie des Sciences de Berlin, XIV (1765), 154-193. [8] G.V. Gorr, Precessional motions in rigid body dynamics and the dynamics of systems of coupled rigid bodies, J. Appl. Math. Mech., 67 (2003), 511-523. [9] G.V. Gorr, L.V. Kudryashova, and L.A. Stepanova, "Classical problems in the theory of solid bodies. Their development and current state," Naukova dumka, Kiev, 1978. [10] G.V. Gorr and V.N. Rubanovskii, A new class of motions of a system of heavy freely connected rigid bodies, J. Appl. Math. Mech., 52 (1988), 551-555. [11] P.V. Kharlamov, The equations of motion of a system of rigid bodies, Mekh. Tverd. Tela, 4 (1972), 52-73. [12] P.V. Kharlamov, Some classes of exact solutions of the problem of the motion of a system of Lagrange gyroscopes, Mat. Fiz., 32 (1982), 63-76. [13] E.I. Kharlamova, A survey of exact solutions of problems of the motion of systems of coupled rigid bodies, Mekh. Tverd. Tela, 26 (1998), 125-138. [14] S.V. Kovalevskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. [15] J.L. Lagrange, "Mechanique Analitique," Veuve Desaint, Paris, 1815. [16] E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about Fixed Point," Springer-Verlag, Berlin, 1965. [17] L. Lilov and N. Vasileva, Steady motion of a system of Lagrange gyroscopes with a tree structure, Teoret. Prilozhna Mekh., XV (1984), 24-34. [18] A.Ya. Savchenko and M.E. Lesina, Particular solution for motion equations of Lagrange gyroscopes system, Mekh. Tverd. Tela, 5 (1973), 27-30. [19] N. Sreenath, Y.G. Oh, P.S. Krishnaprasad, and J.E. Marsden, The dynamics of coupled planar rigid bodies. Part I: Reduction, equilibria and stability, Dynam. Stability Systems, 3 (1988), 25-49. [20] V.A. Steklov, A certain case of motion of a heavy rigid body having a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Lyubit. Estestvozn., 8 (1896), 19-21. [21] J. Wittenburg, "Dynamics of Multibody Systems," Springer-Verlag, Berlin, 2008.
 [1] Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223 [2] Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665 [3] Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893 [4] Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021036 [5] Banavara N. Shashikanth. Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame. Journal of Geometric Mechanics, 2020, 12 (1) : 25-52. doi: 10.3934/jgm.2020003 [6] Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373 [7] Nicolai Sætran, Antonella Zanna. Chains of rigid bodies and their numerical simulation by local frame methods. Journal of Computational Dynamics, 2019, 6 (2) : 409-427. doi: 10.3934/jcd.2019021 [8] Giulio G. Giusteri, Alfredo Marzocchi, Alessandro Musesti. Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2145-2157. doi: 10.3934/dcdsb.2014.19.2145 [9] Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065 [10] Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure and Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881 [11] Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007 [12] Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure and Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23 [13] Bernard Ducomet, Šárka Nečasová. On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1193-1213. doi: 10.3934/dcdss.2013.6.1193 [14] Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417 [15] Liangming Chen, Ming Cao, Chuanjiang Li. Bearing rigidity and formation stabilization for multiple rigid bodies in $SE(3)$. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 257-267. doi: 10.3934/naco.2019017 [16] Anton S. Zadorin. Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1567-1580. doi: 10.3934/cpaa.2022030 [17] Juntang Ding, Chenyu Dong. Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021222 [18] Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control and Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743 [19] Matteo Focardi, Paolo Maria Mariano. Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 109-130. doi: 10.3934/dcdsb.2009.11.109 [20] Tomáš Roubíček, Giuseppe Tomassetti. Dynamics of charged elastic bodies under diffusion at large strains. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1415-1437. doi: 10.3934/dcdsb.2019234

Impact Factor: