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Regul. Chaotic Dyn., 2013, Volume 18, Issue 4, Pages 329–343 (Mi rcd115)  

This article is cited in 21 scientific papers (total in 21 papers)

The Euler–Jacobi–Lie Integrability Theorem

Valery V. Kozlov

V. A. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of $n$ differential equations is proved, which admits $n-2$ independent symmetry fields and an invariant volume $n$-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.

Keywords: symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics


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MSC: 34C14
Received: 05.07.2012

Citation: Valery V. Kozlov, “The Euler–Jacobi–Lie Integrability Theorem”, Regul. Chaotic Dyn., 18:4 (2013), 329–343

Citation in format AMSBIB
\by Valery V. Kozlov
\paper The Euler--Jacobi--Lie Integrability Theorem
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 4
\pages 329--343

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    2. A. V. Tsyganov, “O share Chaplygina v absolyutnom prostranstve”, Nelineinaya dinam., 9:4 (2013), 711–719  mathnet
    3. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika trekh vikhreistochnikov”, Nelineinaya dinam., 10:3 (2014), 319–327  mathnet
    4. Andrey V. Tsiganov, “On the Lie Integrability Theorem for the Chaplygin Ball”, Regul. Chaotic Dyn., 19:2 (2014), 185–197  mathnet  crossref  mathscinet  zmath
    5. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside”, Regul. Chaotic Dyn., 19:2 (2014), 198–213  mathnet  crossref  mathscinet  zmath
    6. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Three Vortex Sources”, Regul. Chaotic Dyn., 19:6 (2014), 694–701  mathnet  crossref  mathscinet  zmath
    7. V. V. Kozlov, “Remarks on Integrable Systems”, Regul. Chaotic Dyn., 19:2 (2014), 145–161  mathnet  crossref  mathscinet  zmath  isi  scopus
    8. Andrey V. Tsiganov, “On Integrable Perturbations of Some Nonholonomic Systems”, SIGMA, 11 (2015), 085, 19 pp.  mathnet  crossref
    9. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453  crossref  mathscinet  zmath  isi  scopus
    10. J. F. Carinena, F. Falceto, J. Grabowski, M. F. Ranada, “Geometry of Lie integrability by quadratures”, J. Phys. A, 48:21 (2015), 215206  crossref  mathscinet  zmath  isi  scopus
    11. Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Note on Free Symmetric Rigid Body Motion”, Regul. Chaotic Dyn., 20:3 (2015), 293–308  mathnet  crossref  mathscinet  zmath  adsnasa
    12. S. Rosemann, K. Schoebel, “Open problems in the theory of finite-dimensional integrable systems and related fields”, J. Geom. Phys., 87 (2015), 396–414  crossref  mathscinet  zmath  isi  scopus
    13. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Vortex Sources in a Deformation Flow”, Regul. Chaotic Dyn., 21:3 (2016), 367–376  mathnet  crossref  mathscinet
    14. Yury A. Grigoryev, Alexey P. Sozonov, Andrey V. Tsiganov, “Integrability of Nonholonomic Heisenberg Type Systems”, SIGMA, 12 (2016), 112, 14 pp.  mathnet  crossref
    15. J. F. Carinena, F. Falceto, J. Grabowski, “Solvability of a Lie algebra of vector fields implies their integrability by quadratures”, J. Phys. A, 49:42 (2016), 425202  crossref  mathscinet  zmath  isi  scopus
    16. V. Kozlov, “The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems”, J. Dyn. Control Syst., 22:4 (2016), 713–724  crossref  mathscinet  zmath  isi  scopus
    17. Božidar Jovanović, “Noether symmetries and integrability in time-dependent Hamiltonian mechanics”, Theor. Appl. Mech., 43:2 (2016), 255–273  mathnet  crossref
    18. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika sanei Chaplygina na tsilindre”, Nelineinaya dinam., 12:4 (2016), 675–687  mathnet  crossref  mathscinet  zmath  elib
    19. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Dynamics of the Chaplygin Sleigh on a Cylinder”, Regul. Chaotic Dyn., 21:1 (2016), 136–146  mathnet  crossref  mathscinet  zmath  elib
    20. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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