Routh method of reduction for dynamical systems with nonstandard Lagrangians on time scales

Song, Jihua; Zhang, Yi

doi:10.1007/s12648-019-01491-x

Cited by 5 publications

(5 citation statements)

References 31 publications

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“…The former is typically expressed as the difference between terms that can be identified as the kinetic and potential energy, while the latter are entitled non-natural and they can take different forms, such as exponential forms, logarithmic forms, and power-law function, etc. [49][50][51][52][53][54][55][56][57][58][59][60]. The non-standard Lagrangians play a significant role in many kinds of equations, such as the nonlinear second-order Riccati equations and the nonlinear differential equation with Lienard type, and the dissipative systems [49][50][51][52][53][54][55][56][57][58][59][60].…”

Section: Discussionmentioning

confidence: 99%

“…[49][50][51][52][53][54][55][56][57][58][59][60]. The non-standard Lagrangians play a significant role in many kinds of equations, such as the nonlinear second-order Riccati equations and the nonlinear differential equation with Lienard type, and the dissipative systems [49][50][51][52][53][54][55][56][57][58][59][60]. Additionally, for all of the equations, standard Lagrangians become invalid.…”

Section: Discussionmentioning

confidence: 99%

“…However, the problem of the exploration and the application of non-standard Lagrangians are still open, and demand lots of deep research. In the past several years, many scientists have made many attempts and great success for studying and applications of non-standard Lagrangians in various fields [49][50][51][52][53][54][55][56][57][58][59][60]. All of the Lagrangians established in this paper are evidently not the standard Lagrangians, because no obvious identification of the kinetic and potential energy terms can be made.…”

Section: Discussionmentioning

confidence: 99%

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Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

et al. 2020

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It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

et al. 2020

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“…Symmetry and conserved quantity have always been a hot spot in analytical mechanics [6][7][8] . Recently, Zhang and his collaborators have studied Noether theorems for systems with non-standard Lagrangians [9,10] , non-standard Hamiltonians [11] , non-standard Birkhoffians [12] , and Lie symmetry [13,14] , Mei symmetry [15] , first integral and method of reduction [16,17] . There have been some results about nonlinear dynamical equations and their symmetries with non-standard Lagrangians [18][19][20][21][22][23][24][25][26] , but canonical transformation and Poisson theory for non-standard Lagrangian dynamics have not been involved.…”

Section: Introductionmentioning

confidence: 99%

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

ZHU,

ZHANG

2024

Wuhan Univ. J. Nat. Sci.

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The canonical transformation and Poisson theory of dynamical systems with exponential, power-law, and logarithmic non-standard Lagrangians are studied, respectively. The criterion equations of canonical transformation are established, and four basic forms of canonical transformations are given. The dynamic equations with non-standard Lagrangians admit Lie algebraic structure. From this, we establish the Poisson theory, which makes it possible to find new conservation laws through known conserved quantities. Some examples are put forward to demonstrate the use of the theory and verify its effectiveness.

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“…Bartosiewicz and his co-workers [24] studied the time-scale second Euler-Lagrange equation, and derived Noether 􀆳 s conserved quantities on time scales by using this equation. Subsequently, many scholars [25][26][27][28][29][30][31][32][33][34] began to carry out a series of studies on time-scale the variational principle for constrained mechanical systems and Noether 􀆳 s theorem of different dynamic systems. However, on a time scale, there are few studies on Lie symmetry in mechanical systems.…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

CHEN

Zhang

2022

Wuhan Univ. J. Nat. Sci.

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The Lie theorem for Birkhoffian systems with time-scale nonshifted variational problems are studied, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system. First, the time-scale nonshifted generalized Pfaff-Birkhoff principle is established, and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced. Afterwards, in the time-scale nonshifted variational problems, by introducing the infinitesimal transformations, Lie symmetry for free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system are defined respectively. Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved. In the end, three examples are given to explain the applications for the results.

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Routh method of reduction for dynamical systems with nonstandard Lagrangians on time scales

Cited by 5 publications

References 31 publications

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

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