Next Article in Journal
Image Enhancement Using Modified Histogram and Log-Exp Transformation
Next Article in Special Issue
Symmetry in Complex Systems
Previous Article in Journal
Refinement Asymptotic Formulas of Eigenvalues and Eigenfunctions of a Fourth Order Linear Differential Operator with Transmission Condition and Discontinuous Weight Function
Previous Article in Special Issue
Time-Fractional Heat Conduction in Two Joint Half-Planes
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Mei Symmetry and Invariants of Quasi-Fractional Dynamical Systems with Non-Standard Lagrangians

1
College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
2
College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(8), 1061; https://doi.org/10.3390/sym11081061
Submission received: 15 July 2019 / Revised: 14 August 2019 / Accepted: 15 August 2019 / Published: 18 August 2019
(This article belongs to the Special Issue Symmetry in Complex Systems)

Abstract

:
Non-standard Lagrangians play an important role in the systems of non-conservative dynamics or nonlinear differential equations, quantum field theories, etc. This paper deals with quasi-fractional dynamical systems from exponential non-standard Lagrangians and power-law non-standard Lagrangians. Firstly, the definition, criterion, and corresponding new conserved quantity of Mei symmetry in this system are presented and studied. Secondly, considering that a small disturbance is applied on the system, the differential equations of the disturbed motion are established, the definition of Mei symmetry and corresponding criterion are given, and the new adiabatic invariants led by Mei symmetry are proposed and proved. Examples also show the validity of the results.

1. Introduction

The study of symmetry and invariants for non-conservative or nonlinear dynamics is of great significance. It is also a frontier research field of analytical mechanics. In a classical sense, the symmetries we refer to mainly include Noether symmetries [1] and Lie symmetries [2,3]. Noether symmetry and Lie symmetry are two different symmetries. After infinitesimal transformation, the former means the invariant property of the Hamilton action functional, and the latter means the invariant property of the differential equation. Unlike Noether symmetry or Lie symmetry, Mei proposed a new symmetry called form invariance in 2000 [4]. Form invariance, also known as Mei symmetry, refers to an invariant property, that is, the dynamical functions (such as Lagrangian, Hamiltonian, Birkhoffian, generalized force, etc.) that appear in the dynamical equations of the mechanical system still satisfy the original equations after the infinitesimal transformation. Under certain conditions, symmetry can lead to invariants, which are also called conserved quantities. Noether symmetry, Lie symmetry, and Mei symmetry of dynamical systems described by standard Lagrangian may lead to Noether conserved quantities or Mei conserved quantities [5], etc. Conserved quantities can also be independent of Lagrangian. For example, conserved quantities can be directly constructed from Lie symmetry neither utilizing Lagrangian nor Hamiltonian, or can be formulated for systems of differential equations by using symmetries and adjoint symmetries together regardless of the existence of a Lagrangian, see [5,6,7,8] and references therein. So far, much progress has been made in the study of the symmetries and corresponding invariants [9,10,11,12,13,14,15,16,17,18,19,20,21,22]. However, there are few reports on the symmetries and invariants of dynamical systems based on non-standard Lagrangians.
The concept of non-standard Lagrangian was first mentioned in Arnold’s works in 1978 as non-natural Lagrangian [23], but it was ignored due to the lack of Hamilton form corresponding to it. Until 1984, when discussing the region adaptability of classical theories in Yang–Mills quantum field theory [24], it was found that non-standard Lagrangians were directly related to the color constraint problem, which led to their renewed attention. The advantage of non-standard Lagrangians is that it can better describe nonlinear problems and plays an important role in non-conservative systems, dissipative systems, quantum field theory, etc. [25,26,27,28,29,30,31,32,33,34].
Fractional calculus can better describe natural phenomena and engineering problems [35,36,37]. Since Riewe [38,39] introduced fractional calculus into the modeling of non-conservative systems, fractional Lagrangian mechanics, fractional Hamiltonian mechanics and fractional Birkhoffian mechanics have been proposed and studied, and important progress has been made in fractional dynamics modeling, analysis, and calculation, see for example [40,41,42,43,44,45,46,47] and references therein. In 2005, El-Nabulsi proposed the fractional action-like variational approach to study non-conservative dynamical problem, in which the action is constructed by using the Riemann-Liouville definition of fractional integral [48,49], and extended it to the case of non-standard Lagrangians [28,50]. Considering the characteristic of fractional action-like variational approach, we call the non-conservative model obtained in this way as quasi-fractional order dynamical system. Here, we propose and study Mei symmetry and its invariants for the quasi-fractional order dynamical system with non-standard Lagrangians. New conserved quantities and new adiabatic invariants are derived from Mei symmetry of the quasi-fractional dynamical systems.

2. Mei Symmetry and Invariants of Quasi-Fractional Dynamical System Based on Exponential Lagrangians

For the quasi-fractional dynamical system whose action functional depends on exponential Lagrangian, the Euler–Lagrange equations that are derived in Appendix A can be expressed as
( t τ ) α 1 exp L ( L q s d d τ L q ˙ s L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = 0 , ( s = 1 , 2 , , n ) ,
where q s ( s = 1 , 2 , , n ) are the generalized coordinates, L = L ( τ , q s , q ˙ s ) is the standard Lagrangian, 0 < α 1 , τ is the intrinsic time, t is the observer time, and τ is not equal to t .
Let us introduce the infinitesimal transformations as
τ * = τ + ε ς 0 ( τ , q k , q ˙ k ) , q s * ( τ * ) = q s ( τ ) + ε ξ s ( τ , q k , q ˙ k ) , ( s = 1 , 2 , , n ; k = 1 , 2 , , n ) ,
where ε is a small parameter, ς 0 and ξ s are the infinitesimals. After the transformation of Equation (2), exp L is transformed into the following form
exp L = exp L ( τ * , q s * , d q s * d τ * ) = exp L ( τ , q s , q ˙ s ) + ε X ( 1 ) ( exp L ) + O ( ε 2 ) ,
where X ( 1 ) is the first extension of the infinitesimal generator X , that is [4]
X = ς 0 τ + ξ s q s ,    X ( 1 ) = ς 0 τ + ξ s q s + ( ξ ˙ s q ˙ s ς ˙ 0 ) q ˙ s .
If L is replaced with L * , Equation (1) still holds, namely
( t τ ) α 1 exp L ( L q s d d τ L q ˙ s L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = 0 , ( s = 1 , 2 , , n ) ,
then this invariance is called Mei symmetry of quasi-fractional dynamical system (1). Substituting the formula (3) into Equation (5), and considering Equation (1), we have
( t τ ) α 1 exp L ( X ( 1 ) ( L ) q s d d τ X ( 1 ) ( L ) q ˙ s X ( 1 ) ( L ) q ˙ s d L d τ L q ˙ s d X ( 1 ) ( L ) d τ + α 1 t τ X ( 1 ) ( L ) q ˙ s ) = 0 ,    ( s = 1 , 2 , , n ) .
Equation (6) is the criterion for Mei symmetry of system (1).
Theorem 1.
For the quasi-fractional dynamical system (1), if there is a gauge function G = G ( τ , q k , q ˙ k ) such that the structural equation
( 1 α t τ ς 0 + ς ˙ 0 ) X ( 1 ) ( exp L ) ( t τ ) α 1 + X ( 1 ) [ X ( 1 ) ( exp L ) ] ( t τ ) α 1 + G ˙ = 0
holds, the Mei symmetry directly leads to the new conserved quantity
I 0 = ( t τ ) α 1 X ( 1 ) ( exp L ) ς 0 + ( t τ ) α 1 X ( 1 ) ( exp L ) q ˙ s ( ξ s q ˙ s ς 0 ) + G = const .
Proof. 
d I 0 d τ = 1 α t τ ( t τ ) α 1 X ( 1 ) ( exp L ) ς 0 + ( t τ ) α 1 d X ( 1 ) ( exp L ) d τ ς 0 + ( t τ ) α 1 X ( 1 ) ( exp L ) ς ˙ 0 + 1 α t τ ( t τ ) α 1 X ( 1 ) ( exp L ) q ˙ s ( ξ s q ˙ s ς 0 ) + ( t τ ) α 1 d d τ X ( 1 ) ( exp L ) q ˙ s ( ξ s q ˙ s ς 0 ) + ( t τ ) α 1 X ( 1 ) ( exp L ) q ˙ s ( ξ ˙ s q ˙ s ς ˙ 0 q ¨ s ς 0 ) ( t τ ) α 1 X ( 1 ) ( exp L ) ( 1 α t τ ς 0 + ς ˙ 0 ) ( t τ ) α 1 X ( 1 ) [ X ( 1 ) ( exp L ) ] = [ X ( 1 ) ( L ) q s + X ( 1 ) ( L ) q ˙ s d L d τ + L q ˙ s d X ( 1 ) ( L ) d τ + d d τ X ( 1 ) ( L ) q ˙ s + 1 α t τ X ( 1 ) ( L ) q ˙ s ] × × ( ξ s q ˙ s ς 0 ) ( t τ ) α 1 exp L + ( 1 α t τ ς 0 + ς ˙ 0 ) X ( 1 ) ( exp L ) ( t τ ) α 1 + X ( 1 ) [ X ( 1 ) ( exp L ) ] ( t τ ) α 1 + ( L q s + d d τ L q ˙ s + L q ˙ s d L d τ + 1 α t τ L q ˙ s ) × × ( ξ s q ˙ s ς 0 ) ( t τ ) α 1 X ( 1 ) ( exp L ) + G ˙ .
Substituting Equations (1) and (6) into the formula (9), and using Equation (7), we obtain
d I 0 d τ = { ( 1 α t τ ς 0 + ς ˙ 0 ) X ( 1 ) ( exp L ) + X ( 1 ) [ X ( 1 ) ( exp L ) ] } ( t τ ) α 1 + G ˙ = 0 .
Thus, we get the desired result. □
The new conserved quantity (8) is called Mei conserved quantity. Since the system is not disturbed, it is an exact invariant. However, in nature and engineering, it is often affected by disturbing forces. If the system is affected by small disturbance υ Q s , its Mei symmetry and the corresponding conserved quantity (8) will change correspondingly. The infinitesimals of transformations (2) without disturbance is denoted as ς 0 0 , ξ s 0 , while the infinitesimals are changed into ς 0 , ξ s when disturbed, and we have
ς 0 = ς 0 0 + υ ς 0 1 + υ 2 ς 0 2 + ,    ξ s = ξ s 0 + υ ξ s 1 + υ 2 ξ s 2 + ,    ( s = 1 , 2 , , n ) .
Meanwhile, we let G 0 represent the gauge function without disturbance, and G represent the gauge function of the disturbed system, which is the small perturbation on the basis of G 0 , i.e.,
G = G 0 + υ G 1 + υ 2 G 2 + .
If a small disturbance υ Q s is applied, Equation (1) is changed to
( t τ ) α 1 exp L ( L q s d d τ L q ˙ s L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = υ Q s .
Accordingly, Equation (6) is changed to
( t τ ) α 1 exp L [ X ( 1 ) ( L ) q s d d τ X ( 1 ) ( L ) q ˙ s X ( 1 ) ( L ) q ˙ s d L d τ L q ˙ s d X ( 1 ) ( L ) d τ + α 1 t τ X ( 1 ) ( L ) q ˙ s ] = υ X ( 1 ) ( Q s ) ,    ( s = 1 , 2 , , n ) .
Substituting the formulae (11) into Equation (14), we get
( t τ ) α 1 υ m exp L [ X m ( 1 ) ( L ) q s d d τ X m ( 1 ) ( L ) q ˙ s X m ( 1 ) ( L ) q ˙ s d L d τ L q ˙ s d X m ( 1 ) ( L ) d τ + α 1 t τ X m ( 1 ) ( L ) q ˙ s ] = υ m + 1 X m ( 1 ) ( Q s ) , ( s = 1 , 2 , , n ) .
where
X ( 1 ) = υ m X m ( 1 ) ,   X m ( 1 ) = ς 0 m τ + ξ s m q s + ( ξ ˙ s m q ˙ s ς ˙ 0 m ) q ˙ s .
As a result, we have
Theorem 2.
If the quasi-fractional dynamical system (1) is disturbed by a small disturbance υ Q s , and there is a gauge function G = G ( τ , q k , q ˙ k ) such that the structural equation
( 1 α t τ ς 0 m + ς ˙ 0 m ) X m ( 1 ) ( exp L ) ( t τ ) α 1 + X m ( 1 ) [ X m ( 1 ) ( exp L ) ] ( t τ ) α 1 + G ˙ m [ X m 1 ( 1 ) ( Q s ) + Q s X m 1 ( 1 ) ( L ) ] ( ξ s m 1 q ˙ s ς 0 m 1 ) = 0 , ( s = 1 , 2 , , n ; m = 0 , 1 , 2 , ) ,
holds, where G = m = 0 z υ m G m and ς 0 1 = ξ s 1 = 0 , the Mei symmetry directly leads to the new adiabatic invariant
I z = m = 0 z υ m [ ( t τ ) α 1 X m ( 1 ) ( exp L ) ς 0 m + ( t τ ) α 1 X m ( 1 ) ( exp L ) q ˙ s ( ξ s m q ˙ s ς 0 m ) + G m ] .
Proof. 
By using Equations (13), (15), and (17), we have
d I z d τ = m = 0 z υ m { 1 α t τ ( t τ ) α 1 X m ( 1 ) ( exp L ) ς 0 m + ( t τ ) α 1 d X m ( 1 ) ( exp L ) d τ ς 0 m + ( t τ ) α 1 X m ( 1 ) ( exp L ) ς ˙ 0 m + 1 α t τ ( t τ ) α 1 X m ( 1 ) ( exp L ) q ˙ s ( ξ s m q ˙ s ς 0 m ) + ( t τ ) α 1 d d τ X m ( 1 ) ( exp L ) q ˙ s ( ξ s m q ˙ s ς 0 m ) + ( t τ ) α 1 X m ( 1 ) ( exp L ) q ˙ s ( ξ ˙ s m q ˙ s ς ˙ 0 m q ¨ s ς 0 m ) + G ˙ m } = m = 0 z υ m { [ X m ( 1 ) ( L ) q s + X m ( 1 ) ( L ) q ˙ s d L d τ + L q ˙ s d X m ( 1 ) ( L ) d τ + d d τ X m ( 1 ) ( L ) q ˙ s + 1 α t τ X m ( 1 ) ( L ) q ˙ s ] × × ( ξ s m q ˙ s ς 0 m ) ( t τ ) α 1 exp L + ( 1 α t τ ς 0 m + ς ˙ 0 m ) X m ( 1 ) ( exp L ) ( t τ ) α 1 + X m ( 1 ) [ X m ( 1 ) ( exp L ) ] ( t τ ) α 1 ( L q s d d τ L q ˙ s L q ˙ s d L d τ 1 α t τ L q ˙ s ) × × ( ξ s m q ˙ s ς 0 m ) ( t τ ) α 1 X m ( 1 ) ( L ) exp L + G ˙ m } = m = 0 z υ m { υ X m ( 1 ) ( Q s ) ( ξ s m q ˙ s ς 0 m ) + X m 1 ( 1 ) ( Q s ) ( ξ s m 1 q ˙ s ς 0 m 1 ) + Q s X m 1 ( 1 ) ( L ) ( ξ s m 1 q ˙ s ς 0 m 1 ) υ Q s X m ( 1 ) ( L ) ( ξ s m q ˙ s ς 0 m ) } = υ z + 1 [ X z ( 1 ) ( Q s ) + Q s X z ( 1 ) ( L ) ] ( ξ s z q ˙ s ς 0 z ) .
According to the definition of adiabatic invariant [51], I z is an adiabatic invariant of order z . This completes the proof. □
Example 1.
Considering the nonlinear dynamical system, its action functional based on exponential Lagrangian is
S = 1 Γ ( α ) t 1 t 2 exp [ L ( τ , q s , q ˙ s ) ] ( t τ ) α 1 d τ ,
where L = τ q q ˙ .
Equation (1) gives
( t τ ) α 1 exp ( τ q q ˙ ) [ τ q ( α 1 t τ q q ˙ τ q ˙ 2 τ q q ¨ ) q ] = 0 .
By calculation, we have
X 0 ( 1 ) ( L ) = q q ˙ ς 0 0 + τ q ˙ ξ 0 + τ q ( ξ ˙ 0 q ˙ ς ˙ 0 0 ) ,
X 0 ( 1 ) ( exp L ) = exp ( τ q q ˙ ) [ q q ˙ ς 0 0 + τ q ˙ ξ 0 + τ q ( ξ ˙ 0 q ˙ ς ˙ 0 0 ) ] .
If we take
ς 0 0 = τ , ξ 0 = 1 q ,
then we have
X 0 ( 1 ) ( L ) = 0 ,    X 0 ( 1 ) ( exp L ) = 0 .
According to the criterion (6), the infinitesimals (24) correspond to Mei symmetry. Substituting (24) into Equation (7), we get
G 0 = τ exp ( τ q q ˙ ) ( t τ ) α 1 .
From Theorem 1, we have
I 0 = τ exp ( τ q q ˙ ) ( t τ ) α 1 = const .
Let the small disturbance be
υ Q = υ q q ˙ exp ( q 2 / 2 ) .
The differential equation of the disturbed motion is
( t τ ) α 1 exp ( τ q q ˙ ) [ τ q ( α 1 t τ q q ˙ τ q ˙ 2 τ q q ¨ ) q ] = υ q q ˙ exp ( q 2 / 2 ) .
Take
ς 0 1 = τ ,    ξ 1 = 1 q ,
then we have
X 1 ( 1 ) ( L ) = 0 ,    X 1 ( 1 ) ( exp L ) = 0 ,    X 0 ( 1 ) ( Q ) = X 1 ( 1 ) ( Q ) = 0 .
According to the criterion (15), the infinitesimals (30) correspond to Mei symmetry. Substituting (30) into Equation (17), we have
G 1 = τ exp ( τ q q ˙ ) ( t τ ) α 1 + υ exp ( q 2 / 2 ) d q .
By Theorem 2, we obtain
I 1 = τ exp ( τ q q ˙ ) ( t τ ) α 1 + υ [ τ exp ( τ q q ˙ ) ( t τ ) α 1 + υ exp ( q 2 / 2 ) d q ] .
The formula (33) is an adiabatic invariant led by Mei symmetry.

3. Mei Symmetry and Invariants of Quasi-Fractional Dynamical System Based on Power-Law Lagrangians

For the quasi-fractional dynamical system whose action functional depends on power-law Lagrangian, the Euler–Lagrange equations that are derived in Appendix B can be expressed as
( 1 + γ ) ( t τ ) α 1 L γ ( L q s d d τ L q ˙ s γ L L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = 0 , ( s = 1 , 2 , , n ) ,
where γ is not equal to 1 .
After the transformation of (2), L 1 + γ is transformed into the following form
L * 1 + γ = L 1 + γ ( τ * , q s * , d q s * d τ * ) = L 1 + γ ( τ , q s , q ˙ s ) + ε X ( 1 ) ( L 1 + γ ) + O ( ε 2 ) .
If L is replaced with L * , Equation (34) still holds, namely
( 1 + γ ) ( t τ ) α 1 L γ ( L q s d d τ L q ˙ s γ L L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = 0 ,    ( s = 1 , 2 , , n ) .
then this invariance is called Mei symmetry of quasi-fractional dynamical system (34). Substituting the formula (35) into Equation (36), and considering Equation (34), we have
( 1 + γ ) ( t τ ) α 1 L γ [ X ( 1 ) ( L ) q s d d τ X ( 1 ) ( L ) q ˙ s γ L d L d τ X ( 1 ) ( L ) q ˙ s γ L L q ˙ s d X ( 1 ) ( L ) d τ + γ L 2 L q ˙ s d L d τ X ( 1 ) ( L ) + α 1 t τ X ( 1 ) ( L ) q ˙ s ] = 0 ,    ( s = 1 , 2 , , n ) .
Equation (37) is the criterion for Mei symmetry of system (34). Hence, we have
Theorem 3.
For the quasi-fractional dynamical system (34), if there is a gauge function G = G ( τ , q k , q ˙ k ) such that the structural equation
( 1 α t τ ς 0 + ς ˙ 0 ) X ( 1 ) ( L 1 + γ ) + X ( 1 ) { X ( 1 ) ( L 1 + γ ) } + ( t τ ) 1 α G ˙ = 0
holds, the Mei symmetry directly leads to the new conserved quantity
I 0 = ( t τ ) α 1 X ( 1 ) ( L 1 + γ ) ς 0 + ( t τ ) α 1 X ( 1 ) ( L 1 + γ ) q ˙ s ( ξ s q ˙ s ς 0 ) + G = const .
Proof. 
d I 0 d t = 1 α t τ ( t τ ) α 1 X ( 1 ) ( L 1 + γ ) ς 0 + ( t τ ) α 1 d X ( 1 ) ( L 1 + γ ) d τ ς 0 + ( t τ ) α 1 X ( 1 ) ( L 1 + γ ) ς ˙ 0 + 1 α t τ ( t τ ) α 1 X ( 1 ) ( L 1 + γ ) q ˙ s ( ξ s q ˙ s ς 0 ) + ( t τ ) α 1 d d τ X ( 1 ) ( L 1 + γ ) q ˙ s ( ξ s q ˙ s ς 0 ) + ( t τ ) α 1 X ( 1 ) ( L 1 + γ ) q ˙ s ( ξ ˙ s q ˙ s ς ˙ 0 q ¨ s ς 0 ) + G ˙ = [ X ( 1 ) ( L ) q s + d d τ X ( 1 ) ( L ) q ˙ s + γ L d L d τ X ( 1 ) ( L ) q ˙ s + γ L L q ˙ s d X ( 1 ) ( L ) d τ γ L 2 L q ˙ s d L d τ X ( 1 ) ( L ) + 1 α t τ X ( 1 ) ( L ) q ˙ s ] ( 1 + γ ) ( t τ ) α 1 L γ ( ξ s q ˙ s ς 0 ) + ( 1 α t τ ς 0 + ς ˙ 0 ) X ( 1 ) ( L 1 + γ ) ( t τ ) α 1 + ( L q s + d d τ L q ˙ s + γ L L q ˙ s d L d τ + 1 α t τ L q ˙ s ) ( 1 + γ ) ( t τ ) α 1 X ( 1 ) ( L γ ) ( ξ s q ˙ s ς 0 ) + X ( 1 ) ( L ) X ( 1 ) ( L ) ( 1 + γ ) γ ( t τ ) α 1 L γ 1 + X ( 1 ) [ X ( 1 ) ( L ) ] ( 1 + γ ) ( t τ ) α 1 L γ + G ˙ .
Substituting Equations (34) and (37) into the formula (40), and using Equation (38), we obtain
d I 0 d t = ( 1 α t τ ς 0 + ς ˙ 0 ) X ( 1 ) ( L 1 + γ ) ( t τ ) α 1 + X ( 1 ) [ X ( 1 ) ( L 1 + γ ) ] ( t τ ) α 1 + G ˙ = 0 .
Thus, we get the desired result. □
Mei conserved quantity (39) is an exact invariant for the quasi-fractional dynamical systems (34).
If a small disturbance υ Q s is applied, Equation (34) is changed to
( 1 + γ ) ( t τ ) α 1 L γ ( L q s d d τ L q ˙ s γ L L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = υ Q s ,    ( s = 1 , 2 , , n ) .
Accordingly, Equation (37) is changed to
( 1 + γ ) ( t τ ) α 1 L γ [ X ( 1 ) ( L ) q s d d τ X ( 1 ) ( L ) q ˙ s γ L d L d τ X ( 1 ) ( L ) q ˙ s γ L L q ˙ s d X ( 1 ) ( L ) d τ + γ L 2 L q ˙ s d L d τ X ( 1 ) ( L ) + α 1 t τ X ( 1 ) ( L ) q ˙ s ] = υ X ( 1 ) ( Q s ) ,    ( s = 1 , 2 , , n ) .
Substituting the formulae (11) into Equation (43), we get
( 1 + γ ) ( t τ ) α 1 υ m L γ [ X m ( 1 ) ( L ) q s d d τ X m ( 1 ) ( L ) q ˙ s γ L d L d τ X m ( 1 ) ( L ) q ˙ s γ L L q ˙ s d X m ( 1 ) ( L ) d τ + γ L 2 L q ˙ s d L d τ X m ( 1 ) ( L ) + α 1 t τ X m ( 1 ) ( L ) q ˙ s ] = υ m + 1 X m ( 1 ) ( Q s ) ,    ( s = 1 , 2 , , n ) .
Therefore, we have
Theorem 4.
If the quasi-fractional dynamical system (34) is disturbed by small disturbance υ Q s , and there is a gauge function G = G ( τ , q k , q ˙ k ) such that the structural equation
( 1 α t τ ς 0 m + ς ˙ 0 m ) X m ( 1 ) ( L 1 + γ ) ( t τ ) α 1 + X m ( 1 ) [ X m ( 1 ) ( L 1 + γ ) ] ( t τ ) α 1 X m 1 ( 1 ) ( Q s ) ( ξ s m 1 q ˙ s ς 0 m 1 ) γ L Q s X m 1 ( 1 ) ( L ) ( ξ s m 1 q ˙ s ς 0 m 1 ) + G ˙ m = 0 ,   ( s = 1 , 2 , , n ; m = 0 , 1 , 2 , ) ,
holds, where G = m = 0 z υ m G m and ς 0 1 = ξ s 1 = 0 , the Mei symmetry directly leads to the new adiabatic invariant
I z = m = 0 z υ m [ ( t τ ) α 1 X m ( 1 ) ( L 1 + γ ) ς 0 m + ( t τ ) α 1 q ˙ s X m ( 1 ) ( L 1 + γ ) ( ξ s m q ˙ s ς 0 m ) + G m ] .
Proof. 
By using Equations (42), (44), and (45), we have
d I z d τ = m = 0 z υ m { 1 α t τ ( t τ ) α 1 X m ( 1 ) ( L 1 + γ ) ς 0 m + ( t τ ) α 1 d X m ( 1 ) ( L 1 + γ ) d τ ς 0 m + ( t τ ) α 1 X m ( 1 ) ( L 1 + γ ) ς ˙ 0 m + 1 α t τ ( t τ ) α 1 X m ( 1 ) ( L 1 + γ ) q ˙ s ( ξ s m q ˙ s ς 0 m ) + ( t τ ) α 1 d d τ X m ( 1 ) ( L 1 + γ ) q ˙ s ( ξ s m q ˙ s ς 0 m ) + ( t τ ) α 1 X m ( 1 ) ( L 1 + γ ) q ˙ s ( ξ ˙ s m q ˙ s ς ˙ 0 m q ¨ s ς 0 m ) + G ˙ m } = m = 0 z υ m { [ X m ( 1 ) ( L ) q s + d d τ X m ( 1 ) ( L ) q ˙ s + γ L d L d τ X m ( 1 ) ( L ) q ˙ s + γ L L q ˙ s d X m ( 1 ) ( L ) d τ γ L 2 L q ˙ s d L d τ X m ( 1 ) ( L ) + 1 α t τ X m ( 1 ) ( L ) q ˙ s ] ( 1 + γ ) ( t τ ) α 1 L γ ( ξ s m q ˙ s ς 0 m ) + ( L q s + d d τ L q ˙ s + γ L L q ˙ s d L d τ + 1 α t τ L q ˙ s ) ( 1 + γ ) ( t τ ) α 1 X m ( 1 ) ( L γ ) ( ξ s m q ˙ s ς 0 m ) + ( 1 α t τ ς 0 m + ς ˙ 0 m ) X m ( 1 ) ( L 1 + γ ) ( t τ ) α 1 + X m ( 1 ) ( L ) X m ( 1 ) ( L ) ( 1 + γ ) γ ( t τ ) α 1 L γ 1 + X m ( 1 ) [ X m ( 1 ) ( L ) ] ( 1 + γ ) ( t τ ) α 1 L γ + G ˙ m } = m = 0 z υ m { υ X m ( 1 ) ( Q s ) ( ξ s m q ˙ s ς 0 m ) υ γ L Q s X m ( 1 ) ( L ) ( ξ s m q ˙ s ς 0 m ) + X m 1 ( 1 ) ( Q s ) ( ξ s m 1 q ˙ s ς 0 m 1 ) + γ L Q s X m 1 ( 1 ) ( L ) ( ξ s m 1 q ˙ s ς 0 m 1 ) } = υ z + 1 [ X z ( 1 ) ( Q s ) + γ L Q s X z ( 1 ) ( L ) ] ( ξ s z q ˙ s ς 0 z ) .
According to the definition of adiabatic invariant [51], I z is an adiabatic invariant of order z . So that ends the proof. □
Example 2.
Considering the nonconservative dynamical system, its action functional based on power-law Lagrangian is [50]
A = 1 Γ ( α ) t 1 t 2 [ L 1 + γ ( τ , q s , q ˙ s ) ] ( t τ ) α 1 d τ ,
where L = q ˙ q ( τ t ) , γ = 1 .
Equation (34) gives
2 ( t τ ) α 1 [ q ¨ + α 1 t τ q ˙ + ( ( τ t ) 2 + α ) q ] = 0 .
By calculation, we have
X 0 ( 1 ) ( L ) = q ς 0 0 ( τ t ) ξ 0 0 + ξ ˙ 0 0 q ˙ ς ˙ 0 0 ,
X 0 ( 1 ) ( L 2 ) = 2 L [ q ς 0 0 ( τ t ) ξ 0 0 + ξ ˙ 0 0 q ˙ ς ˙ 0 0 ] .
Let
ς 0 0 = 0 ,    ξ 0 0 = exp [ ( t + τ ) 2 / 2 ] ,
then
X 0 ( 1 ) ( L ) = 0 ,    X 0 ( 1 ) ( L 2 ) = 0 .
According to the criterion (37), the infinitesimals (52) correspond to Mei symmetry. Substituting (52) into Equation (38), we get
G 0 = 2 [ q ˙ q ( τ t ) ] ( t τ ) α 1 exp [ ( τ t ) 2 / 2 ] .
From Theorem 3, we have
I 0 = 2 [ q ˙ q ( τ t ) ] ( t τ ) α 1 exp [ ( τ t ) 2 / 2 ] = const .
Let the small disturbance be
υ Q = υ sin τ exp [ ( τ t ) 2 / 2 ] .
The differential equation of the disturbed motion is
2 ( t τ ) α 1 [ q ¨ + α 1 t τ q ˙ + ( ( τ t ) 2 + α ) q ] = υ sin τ exp [ ( τ t ) 2 / 2 ] .
Take
ς 0 1 = 0 ,    ξ 0 1 = exp [ ( t + τ ) 2 / 2 ] ,
then it is easy to verify
X 1 ( 1 ) ( L ) = 0 ,    X 1 ( 1 ) ( L 2 ) = 0 ,    X 0 ( 1 ) ( Q ) = X 1 ( 1 ) ( Q ) = 0 .
According to the criterion (44), the infinitesimals (58) correspond to Mei symmetry. Substituting (58) into Equation (45), we have
G 1 = 2 [ q ˙ q ( τ t ) ] ( t τ ) α 1 exp [ ( τ t ) 2 / 2 ] υ cos τ .
By Theorem 4, we obtain
I 1 = 2 [ q ˙ q ( τ t ) ] ( t τ ) α 1 exp [ ( τ t ) 2 / 2 ] + υ { 2 [ q ˙ q ( τ t ) ] ( t τ ) α 1 exp [ ( τ t ) 2 / 2 ] υ cos τ } .
Formula (61) is an adiabatic invariant led by Mei symmetry.

4. Conclusions

Symmetry is closely related to invariants, and it is of great significance to find the invariants of complex system dynamics. First, even if the equations of motion are difficult to solve, the existence of some conserved quantity makes it possible to understand the local physical state or dynamical behavior of the system. Secondly, we can reduce the differential equations of motion by using conserved quantities. Thirdly, we can study the motion stability of complex dynamical systems by using conserved quantities. Based on the quasi-fractional dynamical model proposed by El-Nabulsi according to the Riemann–Liouville definition of fractional integral, we studied Mei symmetry and its corresponding invariants of quasi-fractional dynamics system whose action functional is composed of non-standard Lagrangians. The main results of this paper are its four theorems. In this paper, we provided a method to study nonlinear or non-conservative dynamics and obtained new conserved quantities and new adiabatic invariants, and the results are expected to be generalized or applied to the dynamics of constrained systems, such as those of nonholonomic systems.

Author Contributions

The authors equally contributed to this research work.

Funding

This work was supported by the National Natural Science Foundation of China (grant nos. 11,572,212 and 11,272,227).

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Derivation of the Euler–Lagrange Equations for Quasi-Fractional Dynamical System with Exponential Lagrangians

Consider a nonlinear dynamical system whose configuration is determined by n generalized coordinates q s ( s = 1 , 2 , , n ) , its action functional based on exponential Lagrangian is
S = 1 Γ ( α ) t 1 t 2 exp [ L ( τ , q s , q ˙ s ) ] ( t τ ) α 1 d τ .
where L = L ( τ , q s , q ˙ s ) is the standard Lagrangian, 0 < α 1 , τ is the intrinsic time, t is the observer time, and τ is not equal to t .
The isochronous variational principle
δ S = 0 ,
which satisfies the following commutation relation
d δ q s = δ d q s , ( s = 1 , 2 , , n ) ,
and given boundary condition
δ q s | t = t 1 = δ q s | t = t 2 = 0 , ( s = 1 , 2 , , n )
can be called the Hamilton principle of the quasi-fractional dynamical system with exponential Lagrangians.
Expanding the Hamilton principle (A2), we have
0 = δ S = 1 Γ ( α ) t 1 t 2 δ [ exp L ( t τ ) α 1 ] d τ = 1 Γ ( α ) t 1 t 2 ( t τ ) α 1 exp L ( L q s δ q s + L q ˙ s δ q ˙ s ) d τ
Due to
t 1 t 2 ( t τ ) α 1 exp L L q ˙ s δ q ˙ s d τ = [ ( t τ ) α 1 exp L L q ˙ s δ q s ] | t 1 t 2 t 1 t 2 ( t τ ) α 1 exp L ( α 1 t τ L q ˙ s + d L d τ L q ˙ s + d d τ L q ˙ s ) δ q s d τ = t 1 t 2 ( t τ ) α 1 exp L ( α 1 t τ L q ˙ s + d L d τ L q ˙ s + d d τ L q ˙ s ) δ q s d τ .
Substituting the formula (A6) into Equation (A5), we have
1 Γ ( α ) t 1 t 2 ( t τ ) α 1 exp L ( L q s d d τ L q ˙ s L q ˙ s d L d τ + α 1 t τ L q ˙ s ) δ q s d τ = 0 .
Because of the arbitrariness of the interval [ t 1 , t 2 ] and the independence of δ q s ( s = 1 , 2 , , n ) , using the fundamental lemma [23] of the calculus of variations, we get
( t τ ) α 1 exp L ( L q s d d τ L q ˙ s L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = 0 , ( s = 1 , 2 , , n ) .
Equation (A8) can be called the Euler–Lagrange equations for quasi-fractional dynamical system with exponential Lagrangians.

Appendix B. Derivation of the Euler–Lagrange Equations for Quasi-Fractional Dynamical System with Power-Law Lagrangians

Consider a nonlinear dynamical system whose configuration is determined by n generalized coordinates q s ( s = 1 , 2 , , n ) , its action functional based on power-law Lagrangian is
A = 1 Γ ( α ) t 1 t 2 [ L 1 + γ ( τ , q s , q ˙ s ) ] ( t τ ) α 1 d τ
where L = L ( τ , q s , q ˙ s ) is the standard Lagrangian, γ is not equal to 1 , 0 < α 1 , τ is the intrinsic time, t is the observer time, and τ is not equal to t .
The isochronous variational principle
δ A = 0 ,
which satisfies the following commutation relation
d δ q s = δ d q s , ( s = 1 , 2 , , n ) ,
and given boundary condition
δ q s | t = t 1 = δ q s | t = t 2 = 0 , ( s = 1 , 2 , , n )
can be called the Hamilton principle of the quasi-fractional dynamical system with power-law Lagrangians.
Expanding the Hamilton principle (A10), we have
0 = δ A = 1 Γ ( α ) t 1 t 2 δ [ L 1 + γ ( t τ ) α 1 ] d τ = 1 Γ ( α ) t 1 t 2 ( 1 + γ ) ( t τ ) α 1 L γ ( L q s δ q s + L q ˙ s δ q ˙ s ) d τ
Due to
t 1 t 2 ( t τ ) α 1 L γ L q ˙ s δ q ˙ s d τ = [ ( t τ ) α 1 L γ L q ˙ s δ q s ] | t 1 t 2 t 1 t 2 ( t τ ) α 1 L γ ( α 1 t τ L q ˙ s + γ L d L d τ L q ˙ s + d d τ L q ˙ s ) δ q s d τ = t 1 t 2 ( t τ ) α 1 L γ ( α 1 t τ L q ˙ s + γ L d L d τ L q ˙ s + d d τ L q ˙ s ) δ q s d τ .
Substituting the formula (A14) into Equation (A13), we have
1 Γ ( α ) t 1 t 2 ( 1 + γ ) ( t τ ) α 1 L γ ( L q s d d τ L q ˙ s γ L d L d τ L q ˙ s + α 1 t τ L q ˙ s ) δ q s d τ = 0 .
Because of the arbitrariness of the interval [ t 1 , t 2 ] and the independence of δ q s ( s = 1 , 2 , , n ) , using the fundamental lemma [23] of the calculus of variations, we get
( 1 + γ ) ( t τ ) α 1 L γ ( L q s d d τ L q ˙ s γ L L q ˙ s d L d τ + α 1 t τ L q ˙ s ) = 0 , ( s = 1 , 2 , , n ) .
Equation (A16) can be called the Euler–Lagrange equations for quasi-fractional dynamical system with power-law Lagrangians. Equation (A16) is consistent with the results given in [50].

References

  1. Noether, A.E. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Math. Phys. Kl. 1918, 2, 235–257. [Google Scholar]
  2. Lutzky, M. Dynamical symmetries and conserved quantities. J. Phys. A Math. Gen. 1979, 12, 973–981. [Google Scholar] [CrossRef]
  3. Bluman, G.W.; Anco, S.C. Symmetry and Integration Methods for Differential Equations; Springer: New York, NY, USA, 2002. [Google Scholar]
  4. Mei, F.X. Form invariance of Lagrange system. J. Beijing Inst. Technol. 2000, 9, 120–124. [Google Scholar]
  5. Mei, F.X. Symmetries and Conserved Quantities of Constrained Mechanical Systems; Beijing Institute of Technology Press: Beijing, China, 2004. [Google Scholar]
  6. Hojman, S.A. A new conservation law constructed without using either Lagrangians or Hamiltonians. J. Phys. A Math. Gen. 1992, 25, L291–L295. [Google Scholar] [CrossRef]
  7. Ma, W.X. Conservation laws of discrete evolution equations by symmetries and adjoint symmetries. Symmetry 2015, 7, 714–725. [Google Scholar] [CrossRef]
  8. Ma, W.X. Conservation laws by symmetries and adjoint symmetries. Discret. Cont. Dyn. S 2018, 11, 707–721. [Google Scholar] [CrossRef]
  9. Mei, F.X. Advances in the symmetries and conserved quantities of classical constrained systems. Adv. Mech. 2009, 39, 37–43. [Google Scholar]
  10. Galiullin, A.S.; Gafarov, G.G.; Malaishka, R.P.; Khwan, A.M. Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems; UFN: Moscow, Russia, 1997. [Google Scholar]
  11. Mei, F.X. Lie symmetries and conserved quantities of constrained mechanical systems. Acta Mech. 2000, 141, 135–148. [Google Scholar] [CrossRef]
  12. Zhang, Y.; Zhai, X.H. Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn. 2015, 81, 469–480. [Google Scholar] [CrossRef]
  13. Zhai, X.H.; Zhang, Y. Lie symmetry analysis on time scales and its application on mechanical systems. J. Vib. Control 2019, 25, 581–592. [Google Scholar] [CrossRef]
  14. Jia, L.Q.; Wang, X.X.; Zhang, M.L.; Han, Y.L. Special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system. Nonlinear Dyn. 2012, 69, 1807–1812. [Google Scholar] [CrossRef]
  15. Zhang, Y. Noether’s theorem for a time-delayed Birkhoffian system of Herglotz type. Int. J. Non-Linear Mech. 2018, 101, 36–43. [Google Scholar] [CrossRef]
  16. Djukić, D.S. Adiabatic invariants for dynamical systems with one degree of freedom. Int. J. Non-Linear Mech. 1981, 16, 489–498. [Google Scholar] [CrossRef]
  17. Jiang, W.A.; Luo, S.K. A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems. Nonlinear Dyn. 2012, 67, 475–482. [Google Scholar] [CrossRef]
  18. Song, C.J.; Zhang, Y. Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems. Int. J. Non-Linear Mech. 2017, 90, 32–38. [Google Scholar] [CrossRef]
  19. Yang, M.J.; Luo, S.K. Fractional symmetrical perturbation method of finding adiabatic invariants of disturbed dynamical systems. Int. J. Non-Linear Mech. 2018, 101, 16–25. [Google Scholar] [CrossRef]
  20. Zhang, Y.; Wang, X.P. Lie symmetry perturbation and adiabatic invariants for dynamical system with non-standard Lagrangians. Int. J. Non-Linear Mech. 2018, 105, 165–172. [Google Scholar] [CrossRef]
  21. Luo, S.K.; Yang, M.J.; Zhang, X.T.; Dai, Y. Basic theory of fractional Mei symmetrical perturbation and its application. Acta Mech. 2018, 229, 1833–1848. [Google Scholar] [CrossRef]
  22. Zhang, Y.; Zhai, X.H. Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales. Commun. Nonlinear Sci. Numer. Simulat. 2019, 75, 251–261. [Google Scholar] [CrossRef]
  23. Arnold, V.I. Mathematical Methods of Classical Mechanics; Springer: New York, NY, USA, 1978. [Google Scholar]
  24. Alekseev, A.I.; Arbuzov, B.A. Classical Yang-Mills field theory with nonstandard Lagrangians. Theor. Math. Phys. 1984, 59, 372–378. [Google Scholar] [CrossRef]
  25. Musielak, Z.E. Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A Math. Theor. 2008, 41, 055205. [Google Scholar] [CrossRef]
  26. El-Nabulsi, R.A. Nonlinear dynamics with nonstandard Lagrangians. Qual. Theory Dyn. Syst. 2012, 12, 273–291. [Google Scholar] [CrossRef]
  27. El-Nabulsi, R.A. Non-Standard non-local-in-time Lagrangians in classical mechanics. Qual. Theory Dyn. Syst. 2014, 13, 149–160. [Google Scholar] [CrossRef]
  28. El-Nabulsi, R.A. Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent. Comput. Appl. Math. 2014, 33, 163–179. [Google Scholar] [CrossRef]
  29. Dimitrijevic, D.D.; Milosevic, M. About non-standard Lagrangians in cosmology. AIP Conf. Proc. 2012, 1472, 41. [Google Scholar]
  30. Zhang, Y.; Zhou, X.S. Noether theorem and its inverse for nonlinear dynamical systems with non-standard Lagrangians. Nonlinear Dyn. 2016, 84, 1867–1876. [Google Scholar] [CrossRef]
  31. Song, J.; Zhang, Y. Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales. Chin. Phys. B 2017, 26, 201–209. [Google Scholar] [CrossRef]
  32. Song, J.; Zhang, Y. Noether’s theorems for dynamical systems of two kinds of non-standard Hamiltonians. Acta Mech. 2018, 229, 285–297. [Google Scholar] [CrossRef]
  33. Fiori, S. Extended Hamiltonian learning on Riemannian manifolds: Theoretical aspects. IEEE T. Neur. Net. Lear. 2011, 22, 687–700. [Google Scholar] [CrossRef]
  34. Fiori, S. Extended Hamiltonian learning on Riemannian manifolds: Numerical aspects. IEEE T. Neur. Net. Lear. 2012, 23, 7–21. [Google Scholar] [CrossRef]
  35. Oldham, K.B.; Spanier, J. The Fractional Calculus; Academic Press: San Diego, CA, USA, 1974. [Google Scholar]
  36. Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
  37. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier BV: Amsterdam, The Netherland, 2006. [Google Scholar]
  38. Riewe, F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 1996, 53, 1890–1899. [Google Scholar] [CrossRef]
  39. Riewe, F. Mechanics with fractional derivatives. Phys. Rev. E 1997, 55, 3581–3592. [Google Scholar] [CrossRef]
  40. Agrawal, O.P. Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 2002, 272, 368–379. [Google Scholar] [CrossRef]
  41. Baleanu, D.; Trujillo, J.I. A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives. Commun. Nonlinear Sci. Numer. Simulat. 2010, 15, 1111–1115. [Google Scholar] [CrossRef]
  42. Atanacković, T.M.; Konjik, S.; Pilipović, S.; Simić, S. Variational problems with fractional derivatives: Invariance conditions and Noether’s theorem. Nonlinear Anal. Theory 2009, 71, 1504–1517. [Google Scholar] [CrossRef]
  43. Malinowska, A.B.; Torres, D.F.M. Introduction to the Fractional Calculus of Variations; Imperial College Press: London, UK, 2012. [Google Scholar]
  44. Li, M. Three classes of fractional oscillators. Symmetry 2018, 10, 40. [Google Scholar] [CrossRef]
  45. Zhai, X.H.; Zhang, Y. Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay. Commun. Nonlinear Sci. Numer. Simulat. 2016, 36, 81–97. [Google Scholar] [CrossRef]
  46. Yan, B.; Zhang, Y. Noethe’s theorem for fractional Birkhoffian systems of variable order. Acta Mech. 2016, 227, 2439–2449. [Google Scholar] [CrossRef]
  47. Meng, W.; Zeng, B.; Li, S.L. A novel fractional-order grey prediction model and its modeling error analysis. Information 2019, 10, 167. [Google Scholar] [CrossRef]
  48. El-Nabulsi, R.A. A fractional approach to nonconservative Lagrangian dynamical systems. Fizika A 2005, 14, 289–298. [Google Scholar]
  49. El-Nabulsi, R.A.; Torres, D.F.M. Fractional action-like variational problems. J. Math. Phys. 2008, 49, 053521. [Google Scholar] [CrossRef]
  50. El-Nabulsi, R.A. Non-standard fractional Lagrangians. Nonlinear Dyn. 2013, 74, 381–394. [Google Scholar] [CrossRef]
  51. Zhao, Y.Y.; Mei, F.X. Symmetries and Invariants of Mechanical Systems; Science Press: Beijing, China, 1999. [Google Scholar]

Share and Cite

MDPI and ACS Style

Zhang, Y.; Wang, X.-P. Mei Symmetry and Invariants of Quasi-Fractional Dynamical Systems with Non-Standard Lagrangians. Symmetry 2019, 11, 1061. https://doi.org/10.3390/sym11081061

AMA Style

Zhang Y, Wang X-P. Mei Symmetry and Invariants of Quasi-Fractional Dynamical Systems with Non-Standard Lagrangians. Symmetry. 2019; 11(8):1061. https://doi.org/10.3390/sym11081061

Chicago/Turabian Style

Zhang, Yi, and Xue-Ping Wang. 2019. "Mei Symmetry and Invariants of Quasi-Fractional Dynamical Systems with Non-Standard Lagrangians" Symmetry 11, no. 8: 1061. https://doi.org/10.3390/sym11081061

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop