• Previous Article
    Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition
  • PROC Home
  • This Issue
  • Next Article
    Existence of positive solutions for a system of nonlinear second-order integral boundary value problems
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).   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[4]

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

[5]

J.F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems: a Geometric Approach,, Bibliopolis, (2000).   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

T. Courant, Dirac Manifolds,, Ph.D. Thesis, (1987).   Google Scholar

[10]

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

[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, ().   Google Scholar

[12]

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

[13]

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

[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.   Google Scholar

[15]

A. Kirillov, Local Lie algebras,, Uspekhi Mat. Nauk., 31 (1976), 57.   Google Scholar

[16]

P. Libermann and C.M. Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications,, \textbf{35}, 35 (1987).   Google Scholar

[17]

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

[18]

S. Lie, Theorie der Transformationsgruppen I,, Math. Ann., 16 (1880), 441.   Google Scholar

[19]

S. Lie and G. Scheffers, Vorlesungen Über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,, Teubner, (1893).   Google Scholar

[20]

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

[21]

C.M. Marle, Jacobi manifolds and Jacobi bundles, in:, Symplectic geometry, 20 (1991), 227.   Google Scholar

[22]

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

[23]

I. Vaisman, Lectures on the Geometry of Poisson manifolds,, Birkhäuser Verlag, (1994).   Google Scholar

[24]

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

[25]

P. Winternitz, Lie groups and solutions of nonlinear differential equations, in, Nonlinear Phenomena, 189 (1983), 263.   Google Scholar

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).   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[4]

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

[5]

J.F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems: a Geometric Approach,, Bibliopolis, (2000).   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

T. Courant, Dirac Manifolds,, Ph.D. Thesis, (1987).   Google Scholar

[10]

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

[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, ().   Google Scholar

[12]

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

[13]

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

[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.   Google Scholar

[15]

A. Kirillov, Local Lie algebras,, Uspekhi Mat. Nauk., 31 (1976), 57.   Google Scholar

[16]

P. Libermann and C.M. Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications,, \textbf{35}, 35 (1987).   Google Scholar

[17]

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

[18]

S. Lie, Theorie der Transformationsgruppen I,, Math. Ann., 16 (1880), 441.   Google Scholar

[19]

S. Lie and G. Scheffers, Vorlesungen Über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,, Teubner, (1893).   Google Scholar

[20]

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

[21]

C.M. Marle, Jacobi manifolds and Jacobi bundles, in:, Symplectic geometry, 20 (1991), 227.   Google Scholar

[22]

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

[23]

I. Vaisman, Lectures on the Geometry of Poisson manifolds,, Birkhäuser Verlag, (1994).   Google Scholar

[24]

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

[25]

P. Winternitz, Lie groups and solutions of nonlinear differential equations, in, Nonlinear Phenomena, 189 (1983), 263.   Google Scholar

[1]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239

[2]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066

[3]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[4]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453

[5]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[6]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[7]

Giulia Cavagnari, Antonio Marigonda. Measure-theoretic Lie brackets for nonsmooth vector fields. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052

[8]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39

[9]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351

[10]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517

[11]

K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.

[12]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

[13]

Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121

[14]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[15]

Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295

[16]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014

[17]

Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.

[18]

Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004

[19]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977

[20]

Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347

 Impact Factor: 

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]