Non-standard Birkhoffian dynamics and its Noether’s theorems

doi:10.1016/j.cnsns.2020.105435

Communications in Nonlinear Science and Numerical Simulation

Volume 91, December 2020, 105435

https://doi.org/10.1016/j.cnsns.2020.105435 Get rights and content

Abstract

The most common problems in nature are about non-conservative non-linearity. Non-conservative non-linear problems can be studied with variational problems of non-standard Lagrangians. Birkhoffian mechanics, as an extension of Hamiltonian mechanics naturally, is a sign that analytical mechanics has entered a new stage of development. Therefore, the study of dynamics based on non-standard Birkhoffians provides a new idea for solving non-conservative nonlinear dynamics problems. In this paper the dynamics models based on non-standard Birkhoffians, including exponential Birkhoffian, power law Birkhoffian, and logarithm Birkhoffian, are proposed, which are called non-standard Birkhoffian systems. Firstly, the Pfaff-Birkhoff principles with non-standard Birkhoffians are established, the differential equations of motion of non-standard Birkhoffian dynamics are also derived. Secondly, in accordance with the invariance of Pfaff action under the infinitesimal transformations, giving the definitions and criteria of Noether symmetric and quasi-symmetric transformations of non-standard Birkhoffian dynamics. And next, Noether’s theorems for non-standard Birkhoffian dynamics are proved, and the connections between Noether symmetries and conserved quantities of non-standard Birkhoffian dynamics are established; Finally, three examples are given to illustrate the applications of the results.

Introduction

Non-standard Lagrangians, also known as non-natural Lagrangians, which were first introduced by Arnold in 1978 [1], can be used in the modeling of nonlinear non-conservative dynamical systems, such as nonlinear oscillator systems of Liénard type [2], [3], [4] and Riccati type [5], [6], dissipative systems [7], [8], [9], [10], [11], [12], [13] etc. In 1984, non-standard Lagrangians were applied to the Young-Mills field theory to describe the large distance interactions within the scope of the classical theory [14]. Unlike standard Lagrangians, non-standard Lagrangians generally no longer have the form of the difference between kinetic energy and potential energy. In order to deeply explore the nonlinear differential equations of nonlinear oscillators, El-Nabulsi introduced two types of non-standard Lagrangians, namely, the action with an exponential Lagrangian and that with a power law Lagrangian [15], including time-dependent Lagrangian and time-independent Lagrangian, which provides a new perspective for the study of nonlinear dynamics of complex systems. Since then, these two types of non-standard Lagrangian functions have been applied to modeling nonlinear dynamics, dissipative systems, and classical and quantum dynamics, et al. (see e.g. [16], [17], [18], [19], [20], [21] and references therein). When Saha and Talukdar studied the inverse variational problem, they found that the modified Emden equation and the kinetic equation of some damped harmonic oscillator could be derived from logarithmic Lagrangians [8]. On this basis, El-Nabulsi [22] proposed the variational problem of functional function based on general logarithmic Lagrangians and its non-standard Hamiltonian, and established the modified Euler-Lagrange equations and Hamilton canonical equations.

Conserved quantities have always been a hot issue in analytical mechanics because of the fundamental importance of conservation laws in the study of the dynamics of complex systems. First, even if the differential equations of motion are difficult to solve, the existence of a conserved quantity makes it possible for us to understand the local physical state or dynamic behavior of the system. Secondly, the reduction of differential equations of motion can be realized by using conserved quantities. Thirdly, the conserved quantity also plays an important role in the analysis of the motion stability of the dynamic system. The modern method of finding conserved quantities in a mechanical system is to study its dynamical symmetry. There are mainly Noether symmetry [23], [24], [25], Lie symmetry [26], [27], [28] and Mei symmetry [29], [30]. We consider that if the non-standard Lagrangians and the symmetry theory are combined, it is possible to give a general method for finding the conserved quantities of nonlinear non-conservative dynamical systems. It is based on such considerations that in recent years, we have established Noether symmetry, Lie symmetry and Mei symmetry for the dynamic systems with exponential non-standard Lagrangians and power law non-standard Lagrangians, and obtained the Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity of nonlinear dynamics [18], [19], [31], [32], [33]. The author of literature [34] established the Noether theorem of the system with logarithmic Lagrangians.

However, up to now, all the research on the dynamics based on the three non-standard Lagrangians above has not involved the Birkhoffian system. Birkhoffian mechanics is a natural development of Hamiltonian mechanics and a new stage of analytical mechanics [35]. Birkhoffian mechanics is also the most general possible mechanics, which can be applied to hadronic physics, space mechanics, statistical mechanics, biophysics, engineering, etc. Birkhoffian mechanics originated from the Pfaff variational problem put forward by Birkhoff in his work in 1927 [36], and in 1983 Santilli studied Birkhoffian equations, transformation theory of Birkhoffian equations, as well as the generalization of Galileo’s relativity [37]. In the past 30 years, Mei and his collaborators have conducted in-depth and systematic research on Birkhoffian mechanics, including the integration method of Birkhoffian equations, the symmetry and conservation laws of Birkhoffian systems, the inverse problem of Birkhoffian, the motion stability of Birkhoffian equations, the geometric method and the global analysis of Birkhoffian system [38], [39], [40], [41], [42], [43], [44]. In recent years, some new advances have been made in the study of Birkhoffian mechanics [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57].

The motivation of this article is to combine non-standard Lagrangians and Birkhoffian mechanics, to construct the variational principle and kinetic equations of Birkhoffian systems with non-standard Birkhoffians, and to study the Noether symmetry and conserved quantity of the systems, so as to provide a new perspective for the exploration of nonlinear non-conservative dynamical problems by using the advantages of both Birkhoffian mechanics and non-standard Lagrangians. In the text, we will research the Noether symmetry and conserved quantity (SCQ) of dynamical systems based on non-standard Birkhoffians (including exponential Birkhoffian, power law Birkhoffian, and logarithm Birkhoffian). We put dynamical systems based on non-standard Birkhoffians collectively referred to as non-standard Birkhoffian system. First, the Pfaff-Birkhoff principles for non-standard Birkhoffian systems are established, in the meantime, the corresponding Birkhoffian equations are derived; Secondly, the total variational formulas of the Pfaff actions for non-standard Birkhoffian systems are established, also giving the definitions and criteria of Noether symmetric transformations (NST) and Noether quasi-symmetric transformations (NQST) for non-standard Birkhoffian systems; Finally, Noether’s theorems for non-standard Birkhoffian systems are proved, and applications of the results are illustrated with some examples.

Section snippets

Pfaff-Birkhoff principle and Birkhoffian equations

Integral functional $A_{E} = \int_{t_{1}}^{t_{2}} \exp (R_{μ} {\dot{a}}^{μ} - B) d t$ can be called the Pfaff action based on exponential Birkhoffian, where $B = B (t, a)$ is called Birkhoffian and R_μ are called Birkhoffian functions.

Isochronous variational principle $δ A_{E} = 0$ satisfying the commutation relationships $d δ a^{μ} = δ d a^{μ} (μ = 1, 2, \cdot \cdot \cdot, 2 n)$ and the given boundary conditions $a^{μ} (t) |_{t = t_{1}} = {a_{1}}^{μ}, a^{μ} (t) |_{t = t_{2}} = {a_{2}}^{μ} (μ = 1, 2, \cdot \cdot \cdot, 2 n)$ can be called the Pfaff-Birkhoff principle of non-standard Birkhoffian system based on exponential Birkhoffian. By (2), (3), (4), it is

Pfaff-Birkhoff principle and Birkhoffian equations

Integral functional $A_{P} = \int_{t_{1}}^{t_{2}} (R_{μ} {\dot{a}}^{μ} - B)^{1 + γ} d t$ can be called the Pfaff action based on power law Birkhoffian, where $γ \neq - 1$ .

Isochronous variational principle $δ A_{P} = 0$ satisfying the commutation relationships $d δ a^{μ} = δ d a^{μ} (μ = 1, 2, \cdot \cdot \cdot, 2 n)$ and the given boundary conditions ${a^{μ} (t) |}_{t = t_{1}} = {a_{1}}^{μ}, {a^{μ} (t) |}_{t = t_{2}} = {a_{2}}^{μ} (μ = 1, 2, \cdot \cdot \cdot, 2 n)$ can be called the Pfaff-Birkhoff principle of non-standard Birkhoffian system based on power law Birkhoffian. By (42), (43), (44), we get $\begin{matrix} (1 + γ) {(R_{μ} {\dot{a}}^{μ} - B)}^{γ - 1} {(R_{μ} {\dot{a}}^{μ} - B) [(\frac{\partial R_{ν}}{\partial a^{ρ}} - \frac{\partial R_{ρ}}{\partial a^{ν}}) {\dot{a}}^{ν} - \frac{\partial B}{\partial a^{ρ}} - \frac{\partial R_{ρ}}{\partial t}] - γ \frac{d}{d t} (R_{ν} {\dot{a}}^{ν} - \end{matrix}$

Pfaff-Birkhoff principle and Birkhoffian equations

Integral functional $A_{L} = \int_{t_{1}}^{t_{2}} lo g_{b} (R_{μ} {\dot{a}}^{μ} - B) d t$ can be called the Pfaff action based on logarithm Birkhoffian, where b is the base of the logarithm.

Isochronous variational principle $δ A_{L} = 0$ satisfying the commutation relationships $d δ a^{μ} = δ d a^{μ} (μ = 1, 2, \cdot \cdot \cdot, 2 n)$ and the given boundary conditions ${a^{μ} (t) |}_{t = t_{1}} = {a_{1}}^{μ}, {a^{μ} (t) |}_{t = t_{2}} = {a_{2}}^{μ} (μ = 1, 2, \cdot \cdot \cdot, 2 n)$ can be called the Pfaff-Birkhoff principle of non-standard Birkhoffian system based on logarithm Birkhoffian. By Eqs. (77)-(79), it is easy to get $\frac{1}{(R_{μ} {\dot{a}}^{μ} - B) \ln b} [(\frac{\partial R_{ν}}{\partial a^{ρ}} - \frac{\partial R_{ρ}}{\partial a^{ν}}) {\dot{a}}^{ν} - \partial$

Conclusions

In conclusion, we have proposed in this work three types of non-standard Birkhoffian dynamics which are based on exponential Birkhoffians, power law Birkhoffians, and logarithm Birkhoffians, respectively. The Pfaff-Birkhoff principles, Birkhoffian equations, the Noether symmetry and the Noether conserved quantities for non-standard Birkhoffian systems were investigated. It is reasonable to believe that the main advantage of proposing these three types of Birkhoffian systems lies in a new view

Declaration of Interests Statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement

Lin-Jie Zhang: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Yi Zhang: Conceptualization, Validation, Formal analysis, Visualization, Writing - review & editing, Supervision, Project administration, Funding acquisition.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant Nos. 11972241 and 11572212) and the Natural Science Foundation of Jiangsu Province of China (grant No. BK20191454).

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Research paperNon-standard Birkhoffian dynamics and its Noether’s theorems

Abstract

Introduction

Section snippets

Pfaff-Birkhoff principle and Birkhoffian equations

Pfaff-Birkhoff principle and Birkhoffian equations

Pfaff-Birkhoff principle and Birkhoffian equations

Conclusions

Declaration of Interests Statement

CRediT authorship contribution statement

Acknowledgments

Mathematical methods of classical mechanics

Unusual Liénard-type nonlinear oscillator

Phys Rev E

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

J Math Phys

Nonstandard Lagrangians and branching: the case of some nonlinear Liénard systems

Modern Phys Lett A

Lagrangian formalism for nonlinear second-order riccati systems: one-dimensional integrability and two-dimensional superintegrability

J Math Phys

A simple and unified approach to identify integrable nonlinear oscillators and systems

J Math Phys

A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients

J Phys A: Math Theor

Inverse variational problem for nonstandard Lagrangians

Rep Math Phys

Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

J Phys A: Math Theor

Method to drive Lagrangian and Hamiltonian for a nonlinear dynamical systems with variable coefficients

Chaos Solitons Fractals

General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems

Chaos Solitons Fractals

Nonstandard conserved Hamiltonian structures in dissipative/damped systems: nonlinear generalizations of damped harmonic oscillator

J Math Phys

Classical Yang-Mills field theory with nonstandard Lagrangian

Theor Math Phys

Non-linear dynamics with non-standard Lagrangians

Qual Theory Dyn Syst

Non-standard fractional Lagrangians

Nonlinear Dyn

Fractional variational approach with non-standard power-law degenerate Lagrangians and a generalized derivative operator

Tbilisi J Math

The nonlinear dynamics based on the non-standard Hamiltonians

Nonlinear Dyn

Noether theorem and its inverse for nonlinear dynamical systems with non-standard Lagrangians

Nonlinear Dyn

Lie symmetry perturbation and adiabatic invariants for dynamical system with non-standard Lagrangians

Int J Non-Linear Mech

Non-standard power-law Lagrangians in classical and quantum dynamics

Appl Math Lett

Gravitational field as a pressure force from logarithmic Lagrangians and non-standard Hamiltonians: the case of stellar halo of milky way

Commun Theor Phys

Invariant variations problem

Nachr Akad Wiss Gött Math Phys

Symmetries and conserved quantities of constrained mechanical systems

Int J Dynam Control

Noether symmetries and conserved quantities for fractional Birkhoffian systems

Nonlinear Dyn

Symmetry and integration methods for differential equations

Dynamical symmetries and conserved quantities

J Phys A: Math Gen

Lie symmetries and conserved quantities of constrained mechanical systems

Acta Mech

Research paper
Non-standard Birkhoffian dynamics and its Noether’s theorems