A representation-theoretical approach to higher-dimensional Lie–Hamilton systems: The symplectic Lie algebra <math altimg="si4.svg" display="inline" id="d1e99"><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">p</mi><mrow><mo>(</mo><mn>4</mn><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></math> - INSPIRE

DataBETA

A representation-theoretical approach to higher-dimensional Lie–Hamilton systems: The symplectic Lie algebra $s p (4, R)$

Jun 25, 2024

44 pages

Published in:

Commun.Nonlinear Sci.Numer.Simul. 141 (2025) 108452

Published: Nov 19, 2024

e-Print:

2406.17479 [math-ph]

DOI:

10.1016/j.cnsns.2024.108452 (publication)

View in:

ADS Abstract Service

reference search1 citation

Citations per year

Abstract: (Elsevier B.V.)

A new procedure for the construction of higher-dimensional Lie–Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of intrinsic Lie–Hamilton system is defined, and a sufficiency criterion for this property given. Novel four-dimensional Lie–Hamilton systems arising from the fundamental representation of the symplectic Lie algebra

s p (4, R)

are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra

h_{6}

and the Lorentz Lie algebra

s o (1, 3)

, are also considered in detail. As applications, coupled time-dependent systems which generalize the Bateman oscillator and the one-dimensional Caldirola–Kanai models are constructed, as well as systems depending on a time-dependent electromagnetic field and generalized coupled oscillators. A superposition rule for these systems, exhibiting interesting symmetry properties, is obtained using the coalgebra method.

Note:

44 pages. Some typos and misprints have been corrected. Several comments and an appendix have been added

primary
secondary
Lie systems
Nonlinear differential equations
Nonlinear superposition rules
Symplectic Lie algebra
Lorentz Lie algebra
Two-photon Lie algebra

References(61)

Figures(1)

[1]

S. Lie
,
G. Scheffers

[2]

E. Vessiot

- Annales Fac.Sci.Toulouse 9 (1895) 1-26

[3]

S. Shnider
,
P. Winternitz

- J.Math.Phys. 25 (1984) 3155-3165
•
DOI:
- 10.1063/1.526085

[4]

ON THE INTEGRABILITY OF SYSTEMS OF NONLINEAR ODE'S WITH SUPERPOSITION PRINCIPLES

- J.Math.Phys. 27 (1986) 1215
•
DOI:
- 10.1063/1.527128

[5]

Superposition rules, Lie theorem and partial differential equations

Jose F. Carinena
(
- Zaragoza U.
)
,
Janusz Grabowski
(
- Warsaw, Inst. Math.
)
,
Giuseppe Marmo
(
- Naples U. and
- INFN, Naples
)

- Rept.Math.Phys. 60 (2007) 237-258
•
e-Print:
- math-ph/0610013
•
DOI:
- 10.1016/S0034-4877(07)80137-6

[6]

J. de Lucas
,
C. Sardón

•
DOI:
- 10.1142/q0208

[7]

A. Ballesteros
,
J.F. Cariñena
,
F.J. Herranz
,
J. de Lucas
,
C. Sardón

- J.Phys.A 46 (2013) 285203
•
DOI:
- 10.1088/1751-8113/46/28/285203

[8]

J.F. Cariñena
,
J. de Lucas
,
M.F. Rañada

- Int.J.Geom.Meth.Mod.Phys. 7 (2009) 289-310
•
DOI:
- 10.1142/S0219887810004014

[9]

J.F. Cariñena
,
J. Grabowski
,
G. Marmo

[10]

A. Ballesteros
,
O. Ragnisco

- J.Phys.A 31 (1998) 3791-3813
•
DOI:
- 10.1088/0305-4470/31/16/009

[11]

A. Ballesteros
,
A. Blasco
,
F.J. Herranz
,
J. de Lucas
,
C. Sardón

- J.Diff.Eq. 258 (2015) 2873-2907
•
DOI:
- 10.1016/j.jde.2014.12.031

[12]

A. Blasco
,
F.J. Herranz
,
J. de Lucas
,
C. Sardón

- J.Phys.A 48 (2015) 345202
•
DOI:
- 10.1088/1751-8113/48/34/345202

[13]

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

R. Campoamor-Stursberg

- J.Math.Phys. 59 (2018) 3, 033502
•
DOI:
- 10.1063/1.4989890

[14]

Invariant functions of vector field realizations of Lie algebras and some applications to representation theory and dynamical systems

Rutwig Campoamor-Stursberg

- J.Phys.Conf.Ser. 1071 (2018) 1, 012005
•
DOI:
- 10.1088/1742-6596/1071/1/012005

[15]

Z.-X. Wan

•
DOI:
- 10.1016/C2013-0-05663-0

[16]

É. Cartan

[17]

Differential Forms with Applications to the Physical Sciences

Harley Flanders
(
- Purdue U.
)

[18]

H.J. Sussman

- Trans.Am.Math.Soc. 180 (1973) 171-188
•
DOI:
- 10.2307/1996660

[19]

P. Stefan

- Proc.Lond.Math.Soc. 29 (1974) 699-713
•
DOI:
- 10.1112/plms/s3-29.4.699

[20]

R.S. Palais

•
DOI:
- 10.1090/memo/0022

[21]

E.L. Ince

[22]

L.V. Ovsiannikov

[23]

I. Vaisman

•
DOI:
- 10.1007/978-3-0348-8495-2

[24]

S. Majid

•
DOI:
- 10.1017/CBO9780511613104

[25]

A. Ballesteros
,
R. Campoamor-Stursberg
,
E. Fernández-Saiz
,
F.J. Herranz
,
J. de Lucas

- J.Phys.A 54 (2021) 205202
•
DOI:
- 10.1088/1751-8121/abf1db

1-25 of 61
1
2
3
25 / page