Document Type : Research Paper

Authors

1 Department of Mathematics,Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran.

2 Department of Mathematics, University of Bonab, Tabriz, Iran.

Abstract

In this work, we discuss bi-Hamiltonian structures on a family of integrable systems on 4-dimensional real Lie groups. By constructing the corresponding control matrix  for this family of bi-Hamiltonian structures, we obtain an explicit process for finding  the variables of separation and the separated relations in detail.

Keywords

###### ##### References
[1] J. Abedi-Fardad, A. Rezaei-Aghdam and GH. Haghighatdoost, Integrable and superintegrable Hamiltonian systems with 4-dimensional real Lie algebras as symmetry of the systems, J.Math. Phys., 55 (2014), pp. 1-12.

[2] G. Falqui and M. Pedroni, Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal. Geom., 6 (2003), pp. 139-179.

[3] I.M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct Anal Its Appl, 13 (1979), pp. 248-262.

[4] Gh. Haghighatdoost, H. Abbasi Makrani and R. Mahjoubi, On the cyclic Homology of multiplier Hopf algebras, Sahand Commun. Math. Anal., 9 (2018), pp. 113-128.

[5] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincare, 53 (1990), pp. 35 - 81.

[6] F. Magri and C. Morosi, A Geometrical Characterization of Integrable Hamiltonian Systems through the theory of PoissonNijenhuis manifolds, Dipartimento di Matematica F. Enriques, 1984, Pages 176.

[7] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), pp. 1156-1162.

[8] G. Ovando, Four dimensional symplectic Lie algebras, Beitr. Algebra Geom., 47 (2006), pp. 419-434.

[9] A.V. Tsiganov, New Variables of Separation for the SteklovLyapunov System, SIGMA, Symmetry Integrability Geom. Meth-ods Appl., 8 (2012), pp. 12-14.

[10] A.V. Tsiganov, On bi-hamiltonian structure of some integrable systems on $so*(4)$, J. Nonlinear Math. Phys., 15 (2008), pp. 171-185.

[11] A.V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top, J. Phys. A: Math. Theor., 41 (2008), pp. 1-12.

[12] A.V. Tsiganov, On the two different bi-Hamiltonian structures for the Toda lattice, J. Phys. A, Math. Theor., 40 (2007), pp. 6395-6406.

[13] A.V. Vershilov, A.V. Tsiganov, On bi-Hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion, J. Phys. A, Math. Theor., 42 (2009), pp. 1-12.