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Keywords = Jacobi last multiplier

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30 pages, 423 KiB  
Review
Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics
by José F. Cariñena and José Fernández-Núñez
Symmetry 2021, 13(8), 1413; https://doi.org/10.3390/sym13081413 - 2 Aug 2021
Cited by 16 | Viewed by 3268
Abstract
We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian [...] Read more.
We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given. Full article
9 pages, 232 KiB  
Article
Noether Symmetries Quantization and Superintegrability of Biological Models
by Maria Clara Nucci and Giampaolo Sanchini
Symmetry 2016, 8(12), 155; https://doi.org/10.3390/sym8120155 - 20 Dec 2016
Cited by 9 | Viewed by 4116
Abstract
It is shown that quantization and superintegrability are not concepts that are inherent to classical Physics alone. Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that [...] Read more.
It is shown that quantization and superintegrability are not concepts that are inherent to classical Physics alone. Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that was proposed by Basener and Ross (2005), and that describes the dynamics of growth and sudden decrease in the population of Easter Island. Full article
20 pages, 297 KiB  
Article
Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model
by M.C. Nucci and G. Sanchini
Symmetry 2015, 7(3), 1613-1632; https://doi.org/10.3390/sym7031613 - 8 Sep 2015
Cited by 23 | Viewed by 5612
Abstract
Basener and Ross (2005) proposed a mathematical model that describes the dynamics of growth and sudden decrease in the population of Easter Island. We have applied Lie group analysis to this system and found that it can be integrated by quadrature if the [...] Read more.
Basener and Ross (2005) proposed a mathematical model that describes the dynamics of growth and sudden decrease in the population of Easter Island. We have applied Lie group analysis to this system and found that it can be integrated by quadrature if the involved parameters satisfy certain relationships. We have also discerned hidden linearity. Moreover, we have determined a Jacobi last multiplier and, consequently, a Lagrangian for the general system and have found other cases independently and dependently on symmetry considerations in order to construct a corresponding variational problem, thus enabling us to find conservation laws by means of Noether’s theorem. A comparison with the qualitative analysis given by Basener and Ross is provided. Full article
(This article belongs to the Special Issue Lie Theory and Its Applications)
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