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2015, 2015(special): 605-614. doi: 10.3934/proc.2015.0605

Jacobi--Lie systems: Fundamentals and low-dimensional classification

1. 

Department of Physics, University of Burgos, 09001, Burgos, Spain

2. 

Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warszawa, Poland

3. 

Department of Fundamental Physics, University of Salamanca, Plza. de la Merced s/n, 37.008, Salamanca, Spain

Received  September 2014 Revised  March 2015 Published  November 2015

A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot--Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi--Lie systems. We classify Jacobi--Lie systems on $\mathbb{R}$ and $\mathbb{R}^2$. Our results shall be illustrated through examples of physical and mathematical interest.
Citation: F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
References:
[1]

R. Angelo, E. Duzzioni and A. Ribeiro, Integrability in time-dependent systems with one degree of freedom, J. Phys. A, 45 (2012), 055101.

[2]

A. Ballesteros, A. Blasco, F.J. Herranz, J. de Lucas and C. Sardón, Lie-Hamilton systems on the plane: properties, classification and applications, J. Differential Equations, 258 (2015), 2873-2907.

[3]

A. Ballesteros, J.F. Cariñena, F.J. Herranz, J. de Lucas and C. Sardón, From constants of motion to superposition rules for Lie-Hamilton systems, J. Phys. A, 46 {(2013)}, 285203.

[4]

J.F. Cariñena, J. Grabowski, J. de Lucas and C. Sardón, Dirac-Lie systems and Schwarzian equations, J. Differential Equations, 275 (2014), 2303-2340.

[5]

J.F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems: a Geometric Approach, Bibliopolis, Naples, 2000.

[6]

J.F. Cariñena and J. de Lucas, Lie systems: theory, generalisations, and applications, Dissertationes Math. (Rozprawy Mat.), 479 (2011), 1-162.

[7]

J.F. Cariñena, J. de Lucas and C. Sardón, Lie-Hamilton systems: theory and applications, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1350047.

[8]

J.N. Clelland and P.J. Vassiliou, A solvable string on a Lorentzian surface, Differential Geometry and its Applications, 33 (2014), 177-198.

[9]

T. Courant, Dirac Manifolds, Ph.D. Thesis, University of California, Berkeley, 1987.

[10]

I. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246.

[11]

P.G. Estévez, F.J Herranz, J. de Lucas and C. Sardón, Lie symmetries for Lie systems: applications to systems of ODEs and PDEs, Applied Mathematics and Computation, In press. arXiv:1404.2740.

[12]

Z. Fiala, Evolution equation of Lie-type for finite deformations, time-discrete integration, and incremental methods, Acta Mech., (2015), 17-35.

[13]

R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1169-1177.

[14]

A. González-López, N. Kamran and P.J. Olver, Lie algebras of vector fields in the real plane, Proc. London Math. Soc., 64 (1992), 339-368.

[15]

A. Kirillov, Local Lie algebras, Uspekhi Mat. Nauk., 31 (1976), 57-76.

[16]

P. Libermann and C.M. Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, (1987).

[17]

A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geometry, 12 (1977), 253-300.

[18]

S. Lie, Theorie der Transformationsgruppen I, Math. Ann., 16 (1880), 441-528.

[19]

S. Lie and G. Scheffers, Vorlesungen Über continuierliche Gruppen mit Geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893.

[20]

J. de Lucas and S. Vilariño, $k$-symplectic Lie systems: theory and applications, J. Differential Equations, 258 (2015), 2221-2255.

[21]

C.M. Marle, Jacobi manifolds and Jacobi bundles, in: Symplectic geometry, grupoids and integrable systems, mathematical sciences research institute publications, 20 (1991), 227-246.

[22]

T. Rybicki, On automorphisms of a Jacobi manifold, Univ. Iagel. Acta Math., 38 (2000), 89-98.

[23]

I. Vaisman, Lectures on the Geometry of Poisson manifolds, Birkhäuser Verlag, Basel, 1994.

[24]

N. Weaver, Sub-Riemannian metrics for quantum Heisenberg manifolds, J. Operator Theory, 43 (2000), 223-242.

[25]

P. Winternitz, Lie groups and solutions of nonlinear differential equations, in Nonlinear Phenomena, Lecture Notes in Phys. 189, Springer, Berlin (1983), 263-331.

show all references

References:
[1]

R. Angelo, E. Duzzioni and A. Ribeiro, Integrability in time-dependent systems with one degree of freedom, J. Phys. A, 45 (2012), 055101.

[2]

A. Ballesteros, A. Blasco, F.J. Herranz, J. de Lucas and C. Sardón, Lie-Hamilton systems on the plane: properties, classification and applications, J. Differential Equations, 258 (2015), 2873-2907.

[3]

A. Ballesteros, J.F. Cariñena, F.J. Herranz, J. de Lucas and C. Sardón, From constants of motion to superposition rules for Lie-Hamilton systems, J. Phys. A, 46 {(2013)}, 285203.

[4]

J.F. Cariñena, J. Grabowski, J. de Lucas and C. Sardón, Dirac-Lie systems and Schwarzian equations, J. Differential Equations, 275 (2014), 2303-2340.

[5]

J.F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems: a Geometric Approach, Bibliopolis, Naples, 2000.

[6]

J.F. Cariñena and J. de Lucas, Lie systems: theory, generalisations, and applications, Dissertationes Math. (Rozprawy Mat.), 479 (2011), 1-162.

[7]

J.F. Cariñena, J. de Lucas and C. Sardón, Lie-Hamilton systems: theory and applications, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1350047.

[8]

J.N. Clelland and P.J. Vassiliou, A solvable string on a Lorentzian surface, Differential Geometry and its Applications, 33 (2014), 177-198.

[9]

T. Courant, Dirac Manifolds, Ph.D. Thesis, University of California, Berkeley, 1987.

[10]

I. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246.

[11]

P.G. Estévez, F.J Herranz, J. de Lucas and C. Sardón, Lie symmetries for Lie systems: applications to systems of ODEs and PDEs, Applied Mathematics and Computation, In press. arXiv:1404.2740.

[12]

Z. Fiala, Evolution equation of Lie-type for finite deformations, time-discrete integration, and incremental methods, Acta Mech., (2015), 17-35.

[13]

R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1169-1177.

[14]

A. González-López, N. Kamran and P.J. Olver, Lie algebras of vector fields in the real plane, Proc. London Math. Soc., 64 (1992), 339-368.

[15]

A. Kirillov, Local Lie algebras, Uspekhi Mat. Nauk., 31 (1976), 57-76.

[16]

P. Libermann and C.M. Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, (1987).

[17]

A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geometry, 12 (1977), 253-300.

[18]

S. Lie, Theorie der Transformationsgruppen I, Math. Ann., 16 (1880), 441-528.

[19]

S. Lie and G. Scheffers, Vorlesungen Über continuierliche Gruppen mit Geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893.

[20]

J. de Lucas and S. Vilariño, $k$-symplectic Lie systems: theory and applications, J. Differential Equations, 258 (2015), 2221-2255.

[21]

C.M. Marle, Jacobi manifolds and Jacobi bundles, in: Symplectic geometry, grupoids and integrable systems, mathematical sciences research institute publications, 20 (1991), 227-246.

[22]

T. Rybicki, On automorphisms of a Jacobi manifold, Univ. Iagel. Acta Math., 38 (2000), 89-98.

[23]

I. Vaisman, Lectures on the Geometry of Poisson manifolds, Birkhäuser Verlag, Basel, 1994.

[24]

N. Weaver, Sub-Riemannian metrics for quantum Heisenberg manifolds, J. Operator Theory, 43 (2000), 223-242.

[25]

P. Winternitz, Lie groups and solutions of nonlinear differential equations, in Nonlinear Phenomena, Lecture Notes in Phys. 189, Springer, Berlin (1983), 263-331.

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