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Noether symmetry theory of fractional order constrained Hamiltonian systems based on a fractional factor
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ScienceDirect Karbala International Journal of Modern Science 4 (2018) 180e186 http://www.journals.elsevier.com/karbala-international-journal-of-modern-science/
Noether symmetry theory of fractional order constrained Hamiltonian systems based on a fractional factor Zheng MingLiang School of Mechanical and Engineering, Zhejiang Sci-Tech University, Hangzhou 310018, China Received 25 September 2017; revised 18 December 2017; accepted 24 January 2018 Available online 12 February 2018
Abstract In this paper, we study the Noether Symmetries and conserved quantities of fractional order constrained Hamiltonion systems based on a fractional factor. Firstly, we put forward the calculation method of fractional derivative by the fractional factor, and give the variational problem of fractional systems; Secondly, according to the regular action quantity under the infinitesimal transformation for invariance, we give the definition of Noether symmetric transformation and the criterion equation; Further, according to the relation between symmetries and conserved quantities, we obtain the Noether theorem and its inverse problem. Finally, an example is given to illustrate the application of the result. The research shows that it keeps natural height consistency in the form with the classical integer order constrained mechanical systems by using the derivative definition with fractional factor, the fractional factor can establish the connection between the fractional order systems and the integer order systems. © 2018 The Author. Production and hosting by Elsevier B.V. on behalf of University of Kerbala. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Fractional factor; Fractional order constrained hamiltonian systems; Noether symmetry; Inverse problem
1. Introduction Fractional calculus is almost simultaneous with integer calculus. It is widely used in physics, chemistry, biology, economics and so on. Many scientists and engineers believe that using fractional differential equations to describe the objective world of matter is more true [1]. In 1996, Riewe had studied the dynamic system with nonconservative forces, the fractional EulereLagrange equation and fractional Hamilton equation are established [2,3]. Frederico and Torres introduced the concept of fractional conservation, and
E-mail address: [email protected]. Peer review under responsibility of University of Kerbala.
gave the fractional Nother theorem based on the invariance of the fractional variational problems [4e8]. Atanackovic studied the variational invariance under the definition of Riemann-Liouville fractional derivative, the invariance condition and Noether theorem of the system are presented [9]. In 2005, EI-Nabulsi proposed a new modeling method:class fractional variational method, only one parameterise introduced, the resulting that EulereLagrange equation is simple and similar to the classical equation in form [10,11]. On the basis of EI-Nabulsi's research, Zhang yi studied the fractional variational problem more deeply, and the previous results have been further extended [12,13]. Khalil and Abdeljawad proposed a new definition and basic properties of fractional calculus, which is
https://doi.org/10.1016/j.kijoms.2018.01.005 2405-609X/© 2018 The Author. Production and hosting by Elsevier B.V. on behalf of University of Kerbala. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
consistent with the definition of integer order derivatives, it was called conformable fractional derivatives [14,15]. Fu jingli studied the Noether and Lie symmetries and conserved quantities of fractional order Lagrange and Hamiltonian systems based on joint Caputo derivatives and conformable fractional derivatives [16,17]. Fu jingli studied the Routh equation and the cyclic integral problem of the fractional order Lagrange system based on a fractional factor derivatives, it obtains some new results [18], these results are similar with integer order that it can not easy to obtain in the previous fractional order dynamics. The other research literature about the recent fractional order dynamics systems have [19e21]. Study on the symmetries of fractional nonsingular systems have been obtained some progress. However, under the Legendre transformation, when the singular Lagrange system transits to the phase space and is described by the Hamiltonian systems, there exists an inherent constraint between its canonical variables, which is called the constrained Hamiltonian systems [22]. Many important dynamical systems in reality are the model of constrained Hamiltonian systems [23], such as supersymmetry, supergravity, electromagnetic field, relativistic motion of the particle, superstring and Yang-Mills field etc. However, the research on the variational problem and the symmetries of fractional order constraint Hamiltonian systems is rarely reported, meanwhile, the familiar fractional derivative calculation is complex and involves the operation of special function. It is well known that the Noether theorem plays an important role in classical mechanics, especially in modern science and technology. In this paper, a new definition of fractional derivative by a fractional factor is given, and the Noether symmetry theorem of fractional singular system is further established in phase space. It provides the integral theory of fractional singular systems and verification method for various numerical algorithms 2. Fractional factor and fractional derivative As everyone knows, Riemann-Liouville fractional derivative, Grunwald-Letnikov fractional derivative and Caputo fractional derivative are the integral form of the definition, it has only linear optimality, but its basic properties of calculus with integer order calculus is not a natural consistency. Recently, a novel fractional derivative whose definition and important properties follows [18]. The a order derivative ð0 < a < 1Þ of function y ¼ f ðtÞ, which is defined with fractional factor:
181
f t þ eð1aÞt Da t f ðtÞ df ðtÞ Da ðf Þ ¼ f ðtÞ ¼ lim ¼ Da t/0 Da t da t a
Dt ¼ eð1aÞt Da t dt ¼ eð1aÞt da t ð1Þ Fractional integral based on fractional factor can be used as: Iaab ðf Þ
¼
lim
maxfDti g/0
Zb ¼
n X
Zab f ðxi ÞDa t ¼
i¼1
f ðtÞda t a
eð1aÞt f ðtÞdt
ð2Þ
a
The exchange relations between isochronous variational and fractional order operators, and the fractional differential rule of composite functions are: ddq ð1aÞt dq dDa q ¼ d e ¼ Da dq ¼ eð1aÞt dt dt ð3Þ Da ðfgÞ ¼ Da ðf Þ,g þ Da ðgÞ,f 3. Variational problems and equations of motion The form of fractional order mechanical systems is determine by generalized coordinates qs ðs ¼ 1; 2; :::; nÞ, the Lagrange function of the system is Lðt; q; Da qÞ 0 < a < 1, introduced the generalized momentum pas and fractional order Hamilton function H a are: vL pas ¼ vDa qs ð4Þ H a ðt; q; pa Þ ¼ pas Da qs Lðt; q; Da qÞ If the rank of hessian matrix of L is r < n, that is 2 L ¼ 0, so the systems are a detðhsk Þ ¼ vDa qvs vD a qk fractional singular systems. When the Lagrange describes transition to the Hamiltonian systems, all Da qs ðtÞ can not be solved by Formula (4) as the function of t; q; pa , an inherent constraint exists between regular variables (qs ; pas is not complete independence): 4j ðt; q; pa Þ ¼ 0ðj ¼ 1; 2; :::; n rÞ
ð5Þ
The variational problem of the fractional order Hamiltonian systems can be described as:
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Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
Zb dI ¼ d
Lðt; qs ; Da qs Þe
ð1aÞt
Zb da t ¼ d
a
Lðt; qs ; Da qs Þdt a
Zb ½pas Da qs H a ðt; qs ; pas Þdt
¼d a
dt ¼ 0; dqs jt¼a ¼ dqs jt¼b ¼ 0 ð6Þ The fractional order Hamilton variable principle is:
Zb Da qs dpas þ pas dDa qs a
Zb ¼ a
ð1aÞt
b pas dqs þ
ð10Þ
vH a vH a dqs dpas dt vqs vpas
Z b
a
a
Z b a
8 vH *a > ð1aÞt v4j > _ q ¼ lj ¼ gs ðt; q; pa Þ; > < s vp þ e vpas as > v4j > vH *a > : p_as ¼ eð1aÞt lj ¼ hs ðt; q; pa Þ vqs vqs
vH a vH a dqs dpas dt Da qs dpas þ Da ðpas dqs Þ dqs Da pas vqs vpas
¼ e
¼
Now we take H *a ðt; q; pa Þ ¼ eð1aÞt H a ðt; q; pa Þ, according to the fractional factor calculation method, the differential equations of motion in phase space can be expanded to:
vH a vH a D a qs dpas Da pas dqs dt vpas vqs
ð7Þ
vH a vH a D a qs dpas Da pas dqs dt vpas vqs
¼0
The equations are restricted by the inherent constraint (5), it needs satisfy to dqs ; dpas :
Especially if a ¼ 1, Equation (10) become the motion equation of singular systems with integer order in phase space. 4. Noether symmetry
d4j d4j dqs þ dpas ¼ 0 vqs vqas
ð8Þ
For nonindependence qs ; pas , it can properly chosen lj , then, the canonical equation of fractional order constrained Hamiltonian systems are obtained by the arbitrariness of the integral interval: 8 v4j vH a > > D q ¼ > < a s vpas þ lj vpas ðs ¼ 1; 2; :::; nÞ > v4j vH a > > : Da pas ¼ lj vqs vqs
ð9Þ
Set ε as an infinitesimal parameter, x0 ; xs ; hs is the generator of infinitesimal transformation, the infinitesimal transformations contain time, generalized coordinates and generalized momenta: t* ¼ t þ εx0 ðt; qs ; pas Þ; q*s ¼ qs þ εxs ðt; qs ; pas Þ; p*as ¼ pas þ εhs ðt; qs ; pas Þ
ð11Þ
In phase space, the total variation of the regular action quantity of fractional systems under the infinitesimal transformations is:
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
Zb DS ¼ a
p*as Da q*s
¼ Iaab DL þ L
H t* ; q*s ; p*as da t*
Zb ½pas Da qs H a ðt; qs ; pas Þda t
a
dDt dt
183
a
ð12Þ
vH a vH a vH a dDt ¼ Iaab pas DDa qs þ Dpas Da qs Dt Dqs Dpas þ ½pas Da qs H a dt vt vpas vpas
Be aware: Dt ¼ εx0 dqs ¼ Dqs q_s Dt ¼ εðxs q_s x0 Þ ¼ εxs dpas ¼ Dpas p_as Dt ¼ εðhs p_as x0 Þ ¼ εhs DDa qs ¼ dDa qs þ Daþ1 qs Dt ¼ Da dqs þ Daþ1 qs Dt ¼ εðDa xs q_s Da x0 Þ ð13Þ Below, we introduce the definition and criterion equations of the Noether symmetric transformation for fractional order constrained Hamiltonian systems (Noether identities). Definition: If the regular action quantity of the fractional order constraint systems is the quasi invariant under the infinitesimal transformation (11) in phase space, that is, for each generator x0 ; xs ; hs, it is always established:
DS ¼ Iaab Da ðDGÞ Iaab Lqs dqs þ Lpas dpas ð14Þ v4j v4j ¼ Iaab Da ðεGÞ Iaab lj dqs þ lj dpas vqs vpas
From the above basic Formula (12) about variational of fractional order constraint Hamiltonian systems, then we can obtain the criterion equation of the Noether quasi symmetric transformation: vH vH pas ðDa xs q_s Da x0 Þ þ ðDa qs Þhs x0 xs vt vqs v4j v4j vH hs þ ðpas Da qs HÞx_ 0 lj xs þ lj h vpas vqs vpas s þ Da G ¼ 0 ð15Þ 5. Noether theorem and inverse theorem 5.1. The Noether theorem of fractional order constrained Hamiltonian systems According to the relation between total variation and isochronous variation, the total variation of the regular action quantity of fractional order singular system (12) can also be expressed as:
_ DS ¼ dS þ SDt vH a vH a ¼ Iaab dqs dpas Da qs dpas þ pas dDa qs þ ðpas Da qs H a ÞDt vqs vpas vH a vH a ab a ¼ Ia Da ½pas þ dqs þ ðpas Da qs H ÞDt þ Da qs dpas þ Da ps dqs vqas vqs
The gauge function G ¼ Gðt; qs ; pas Þ, then, the infinitesimal transformation is the Noether quasi symmetric transformation.
ð16Þ
According to Formula (13), so Formula (16) can be simplification to:
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Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
DS ¼ Da pas xs þ ðpas Da qs H Þx0 þ Da qs vH a vH a hs þ Da ps x vpas vqs s Iaab ε
a
ð17Þ Sum deduced from above, we can easily obtain the Noether theorem for fractional order constrained Hamiltonian systems: Theorem 1: If the generator x0 ðt; qs ; pas Þ; xs ðt; qs ; pas Þ, hs ðt; qs ; pas Þ satisfy the criterion Equation (15) of Noether quasi symmetry for fractional order constraints Hamiltonian systems, then the system have the first integral in the following form: I ¼ pas xs þ ðpas Da qs HÞx0 þ G ¼ const
ð18Þ
5.2. Noether inverse theorem for fractional order constrained Hamiltonian systems It is assumed that a fractional order constrained Hamiltonian systems have the number of l linearly independent conserved quantities in the extended phase space: I s ðt; q; pa Þ ¼ constðs ¼ 1; 2:::; lÞ
ð19Þ
For fractional order constrained Hamiltonian systems, any trajectory is satisfied (9), have:
v4j v4j vH a vH a s Da qs lj h þ D a ps lj vpas vpas s vqs vqs s
xs ¼ 0 ð20Þ The both sides of (19) calculation fractional derivatives about time t, then add it up to (20), have: vI s vI s vI s vH a Da t þ Da qs þ Da pas þ Da qs vt vqs vpas vpas a v4j v4j s vH lj hs þ Da ps lj x vpas s vqs vqs s ¼0
s
s
I ¼ pas xs þ Ls xs0 Gs
ð23Þ
So:
xs0
s s s ¼ L1 s I pas xs þ G
ð24Þ
By (22) and (24) expressions, all xss can be found immediately. Also because: dq*s dqs dqs þ εs xss dq
s ¼ da t* da t da t 1 þ εs da xs0 da t da t s s dxs dx q_s 0 ¼ eð1aÞt εs dt dt
DDa qs ¼
ð25Þ Thus the generator hss ðt; q; pa Þ of pas , it can obtained by xs0 ; xss in Formula (25), such find out xs0 ; xss and hss , as long as they are satisfied: v4j s v4j s x q_s xss þ h p_as xs0 ¼ 0 ð26Þ vqs s vpas s Thus, we obtain the inverse theorem of the regular form Noethor theorem for fractional order constrained Hamiltonian systems: Theorem 2:If the number of l linearly independent of the first integral (motion conserved quantities) (19) about the canonical Equation (10) of holonomic fractional order constraint Hamiltonian systems are known, so, the infinitesimal transformation (11) are formed with xs0 ; xss and hss , it is determined by (24), (22) and (25), as long as Formula (26) is also established, under this transformation, the invariance of the regular action quantity of fractional order constrained Hamiltonian Systems must be suitable for Noether quasi symmetry definitions (14). 6. Illustrated example Example The Lagrange function of fractional order system with two degrees of freedom is: L ¼ ðDa q1 Þq2 ðDa q2 Þq1 þ q21 þ q22
ð21Þ
These relations are always satisfied along arbitrary trajectories, so all coefficients of Da pas should be zero, it is: xs ¼ vI s =vpas
Now take: s
ð22Þ
ð27Þ
0 < a < 1 Please study the Noether symmetry of the system and its inverse problem. The generalized momentum and fractional Hamilton functions of the system are: vL vL pa1 ¼ ¼ q2 ; pa2 ¼ ¼ q1 vDa q1 vDa q2 ð28Þ H a ¼ Pa1 Da q1 þ pa2 Da q2 L ¼ q21 q22
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
The rank of hessian matrix of L is r ¼ 0, Thus, there are two inherent constraints between canonical variables: 41 ¼ pa1 q2 ¼ 0; 42 ¼ pa2 þ q1 ¼ 0
ð29Þ
185
The generator determined by the (38) satisfies the constraint Equation (26), therefore, the Noether quasi symmetry corresponding to fractional order constrained Hamiltonian systems is considered.
The stable consistency conditions are bound by constraints, it can obtained:
7. Conclusions
l1 ¼ q2 ; l2 ¼ q1
In order to solve the large number of singular Lagrange functions problem in modern physics and mechanics, in this paper, Noether symmetry theory has been extended to fractional order constrained Hamiltonian systems based on a fractional factor, A series of new results have been given for the first time, the canonical equation, Noether symmetric criterion equation, conserved quantities and inverse theorem are very concise and completely consistent in form with the conclusion of the classic mechanical systems of integer order. When a is equal to 1, all the results of this paper are degenerated to the Noether theorem of integer order constrained Hamilton systems, it shows that the fractional factor is more extensive. Using the research methods of this paper, The Noether symmetry theory of nonholonomic fractional order constrained Hamiltonian systems and singular fractional electromechanical coupled systems can be further studied.
ð30Þ
The quasi Noether criterion Equation (15) under the definition of quasi symmetry, we have: pas ðDa xs q_s Da x0 Þ þ ðDa qs Þhs þ 2qs xs þ þLx_ 0 þ Da G þ l1 ½h1 x2 þ ðq_2 p_a1 Þx0 þ l2 ½h2 þ x1 ðq_1 þ p_a2 Þx0 ¼ 0
ð31Þ
The upper form follows a set of solutions: x0 ¼ 1; x1 ¼ q2 ; x2 ¼ q1 ; h1 ¼ q1 ; h2 ¼ q2 Gðt; q; pa Þ ¼ 0
ð32Þ
The quasi Noether symmetry leads to the corresponding conserved quantity: I ¼ eð1aÞt pa2 q1 eð1aÞt pa1 q2 þ q21 þ q22 ¼ const ð33Þ Secondly, we study the inverse problem of Noether symmetry. Set up fractional order constrained Hamiltonian systems have integral: I ¼ eð1aÞt pa2 q1 eð1aÞt pa1 q2 þ q21 þ q22 ¼ const ð34Þ By Equations (22) And (24) give the infinitesimal generator of the system are satisfied: x1 q_1 x0 ¼ 0; x2 q_2 x0 ¼ 0 x0 ¼ L1 ½I pa1 ðx1 q_1 x0 Þ pa2 ðx2 q_2 x0 Þ þ G ð35Þ Take Gðt; q; pa Þ ¼ 0, then we get a set of Solutions: x0 ¼ 1; x1 ¼ q2 ; x2 ¼ q1
ð36Þ
According to the expression of regular momentum (25), the other generators are obtained: h1 ¼ x2 ; h2 ¼ x1
ð37Þ
Therefore, the infinitesimal transformation generator of the system can be: x0 ¼ 1; x1 ¼ q2 ; x2 ¼ q1 ; h1 ¼ q1 ; h2 ¼ q2
ð38Þ
Conflict of interest None. References [1] W. Sun, H.G. Sun, X.C. Li, Fractional Derivative Modeling of Mechanics and Engineering Problems (In Chinese), Science Press, Beijing, 2010, p. 3. [2] F. Riewe, Phys. Rev. E 53 (2) (1996) 1890. [3] F. Riewe, Phys. Rev. E 55 (3) (1997) 3581. [4] G.S.F. Frederico, D.F.M. Torres, Int. J. Tomogr. Stat. 5 (W07) (2007) 109. [5] G.S.F. Frederico, D.F.M. Torres, Math. Anal. Appl. 334 (2) (2007) 834. [6] G.S.F. Frederico, D.F.M. Torres, Proceed-ings of the 2nd IFAC Workshop on Fractional Deferentia-anon and its Applications (Portugal), 2006, pp. 142e147. [7] G.S.F. Frederico, D.F.M. Torres, Nonlinear Dyn. 53 (3) (2008) 215. [8] A.B. Malinowska, D.F.M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012, p. 11. [9] T.M. Atanackovic, S. Konjik, Nonlinear Ana1y 71 (5/6) (2009) 1504. [10] A.R. EL-nabulsi, Fizika A 14 (4) (2005) 289. [11] A.R. El-nabulsi, Applied Mathematicand Computation 217 (2011) 9492.
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[12] Y. Zhang, J. Zhong. Univ. (Natural Science Edition) (in Chinese) 5 (2) (2013) 45. [13] Z.X. Long, Y. Zhang, J. Zhong. Univ. (Natural Science Edition)(in Chinese) 5 (2) (2013) 51. [14] R. Khalil, H.M. Al, A. Yousef, M.J. Sababheh, J.Comput. Appl. Math. 264 (2014) 65. [15] T. Abdeljawad, J. Comput. Appl. Math. 279 (2015) 57. [16] L.L. Wang, J.L. Fu, Chin. Phys. B 23 (12) (2014) 277. [17] L.L. Wang, J.L. Fu, Chin. Phys. B 25 (1) (2016) 146. [18] J.L. Fu, Chin. Q. Mech. 37 (2016) 252.
[19] X. Min, X.Z. Wei, D.C. Jin, Math. Comput. Simulat. 89 (2013) 1e12. [20] S. Jun, D.C. Jin, Asian J. Contr. 14 (6) (2012) 1690e1697. [21] I.K. Dassios, Math. Meth. Appl. Sci. 40 (17) (2017) 6085e6095. [22] Z.P. Li, Constrained Hamiltonian system and its symmetry (in Chinese), Beijing University of Technology press, Beijing, 1999, p. 25. [23] P.A.M. Dirac, Lecture on Quantum Mechanics, Yeshiva University press, NewYork, 1964, p. 73.
ScienceDirect Karbala International Journal of Modern Science 4 (2018) 180e186 http://www.journals.elsevier.com/karbala-international-journal-of-modern-science/
Noether symmetry theory of fractional order constrained Hamiltonian systems based on a fractional factor Zheng MingLiang School of Mechanical and Engineering, Zhejiang Sci-Tech University, Hangzhou 310018, China Received 25 September 2017; revised 18 December 2017; accepted 24 January 2018 Available online 12 February 2018
Abstract In this paper, we study the Noether Symmetries and conserved quantities of fractional order constrained Hamiltonion systems based on a fractional factor. Firstly, we put forward the calculation method of fractional derivative by the fractional factor, and give the variational problem of fractional systems; Secondly, according to the regular action quantity under the infinitesimal transformation for invariance, we give the definition of Noether symmetric transformation and the criterion equation; Further, according to the relation between symmetries and conserved quantities, we obtain the Noether theorem and its inverse problem. Finally, an example is given to illustrate the application of the result. The research shows that it keeps natural height consistency in the form with the classical integer order constrained mechanical systems by using the derivative definition with fractional factor, the fractional factor can establish the connection between the fractional order systems and the integer order systems. © 2018 The Author. Production and hosting by Elsevier B.V. on behalf of University of Kerbala. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Fractional factor; Fractional order constrained hamiltonian systems; Noether symmetry; Inverse problem
1. Introduction Fractional calculus is almost simultaneous with integer calculus. It is widely used in physics, chemistry, biology, economics and so on. Many scientists and engineers believe that using fractional differential equations to describe the objective world of matter is more true [1]. In 1996, Riewe had studied the dynamic system with nonconservative forces, the fractional EulereLagrange equation and fractional Hamilton equation are established [2,3]. Frederico and Torres introduced the concept of fractional conservation, and
E-mail address: [email protected]. Peer review under responsibility of University of Kerbala.
gave the fractional Nother theorem based on the invariance of the fractional variational problems [4e8]. Atanackovic studied the variational invariance under the definition of Riemann-Liouville fractional derivative, the invariance condition and Noether theorem of the system are presented [9]. In 2005, EI-Nabulsi proposed a new modeling method:class fractional variational method, only one parameterise introduced, the resulting that EulereLagrange equation is simple and similar to the classical equation in form [10,11]. On the basis of EI-Nabulsi's research, Zhang yi studied the fractional variational problem more deeply, and the previous results have been further extended [12,13]. Khalil and Abdeljawad proposed a new definition and basic properties of fractional calculus, which is
https://doi.org/10.1016/j.kijoms.2018.01.005 2405-609X/© 2018 The Author. Production and hosting by Elsevier B.V. on behalf of University of Kerbala. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
consistent with the definition of integer order derivatives, it was called conformable fractional derivatives [14,15]. Fu jingli studied the Noether and Lie symmetries and conserved quantities of fractional order Lagrange and Hamiltonian systems based on joint Caputo derivatives and conformable fractional derivatives [16,17]. Fu jingli studied the Routh equation and the cyclic integral problem of the fractional order Lagrange system based on a fractional factor derivatives, it obtains some new results [18], these results are similar with integer order that it can not easy to obtain in the previous fractional order dynamics. The other research literature about the recent fractional order dynamics systems have [19e21]. Study on the symmetries of fractional nonsingular systems have been obtained some progress. However, under the Legendre transformation, when the singular Lagrange system transits to the phase space and is described by the Hamiltonian systems, there exists an inherent constraint between its canonical variables, which is called the constrained Hamiltonian systems [22]. Many important dynamical systems in reality are the model of constrained Hamiltonian systems [23], such as supersymmetry, supergravity, electromagnetic field, relativistic motion of the particle, superstring and Yang-Mills field etc. However, the research on the variational problem and the symmetries of fractional order constraint Hamiltonian systems is rarely reported, meanwhile, the familiar fractional derivative calculation is complex and involves the operation of special function. It is well known that the Noether theorem plays an important role in classical mechanics, especially in modern science and technology. In this paper, a new definition of fractional derivative by a fractional factor is given, and the Noether symmetry theorem of fractional singular system is further established in phase space. It provides the integral theory of fractional singular systems and verification method for various numerical algorithms 2. Fractional factor and fractional derivative As everyone knows, Riemann-Liouville fractional derivative, Grunwald-Letnikov fractional derivative and Caputo fractional derivative are the integral form of the definition, it has only linear optimality, but its basic properties of calculus with integer order calculus is not a natural consistency. Recently, a novel fractional derivative whose definition and important properties follows [18]. The a order derivative ð0 < a < 1Þ of function y ¼ f ðtÞ, which is defined with fractional factor:
181
f t þ eð1aÞt Da t f ðtÞ df ðtÞ Da ðf Þ ¼ f ðtÞ ¼ lim ¼ Da t/0 Da t da t a
Dt ¼ eð1aÞt Da t dt ¼ eð1aÞt da t ð1Þ Fractional integral based on fractional factor can be used as: Iaab ðf Þ
¼
lim
maxfDti g/0
Zb ¼
n X
Zab f ðxi ÞDa t ¼
i¼1
f ðtÞda t a
eð1aÞt f ðtÞdt
ð2Þ
a
The exchange relations between isochronous variational and fractional order operators, and the fractional differential rule of composite functions are: ddq ð1aÞt dq dDa q ¼ d e ¼ Da dq ¼ eð1aÞt dt dt ð3Þ Da ðfgÞ ¼ Da ðf Þ,g þ Da ðgÞ,f 3. Variational problems and equations of motion The form of fractional order mechanical systems is determine by generalized coordinates qs ðs ¼ 1; 2; :::; nÞ, the Lagrange function of the system is Lðt; q; Da qÞ 0 < a < 1, introduced the generalized momentum pas and fractional order Hamilton function H a are: vL pas ¼ vDa qs ð4Þ H a ðt; q; pa Þ ¼ pas Da qs Lðt; q; Da qÞ If the rank of hessian matrix of L is r < n, that is 2 L ¼ 0, so the systems are a detðhsk Þ ¼ vDa qvs vD a qk fractional singular systems. When the Lagrange describes transition to the Hamiltonian systems, all Da qs ðtÞ can not be solved by Formula (4) as the function of t; q; pa , an inherent constraint exists between regular variables (qs ; pas is not complete independence): 4j ðt; q; pa Þ ¼ 0ðj ¼ 1; 2; :::; n rÞ
ð5Þ
The variational problem of the fractional order Hamiltonian systems can be described as:
182
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
Zb dI ¼ d
Lðt; qs ; Da qs Þe
ð1aÞt
Zb da t ¼ d
a
Lðt; qs ; Da qs Þdt a
Zb ½pas Da qs H a ðt; qs ; pas Þdt
¼d a
dt ¼ 0; dqs jt¼a ¼ dqs jt¼b ¼ 0 ð6Þ The fractional order Hamilton variable principle is:
Zb Da qs dpas þ pas dDa qs a
Zb ¼ a
ð1aÞt
b pas dqs þ
ð10Þ
vH a vH a dqs dpas dt vqs vpas
Z b
a
a
Z b a
8 vH *a > ð1aÞt v4j > _ q ¼ lj ¼ gs ðt; q; pa Þ; > < s vp þ e vpas as > v4j > vH *a > : p_as ¼ eð1aÞt lj ¼ hs ðt; q; pa Þ vqs vqs
vH a vH a dqs dpas dt Da qs dpas þ Da ðpas dqs Þ dqs Da pas vqs vpas
¼ e
¼
Now we take H *a ðt; q; pa Þ ¼ eð1aÞt H a ðt; q; pa Þ, according to the fractional factor calculation method, the differential equations of motion in phase space can be expanded to:
vH a vH a D a qs dpas Da pas dqs dt vpas vqs
ð7Þ
vH a vH a D a qs dpas Da pas dqs dt vpas vqs
¼0
The equations are restricted by the inherent constraint (5), it needs satisfy to dqs ; dpas :
Especially if a ¼ 1, Equation (10) become the motion equation of singular systems with integer order in phase space. 4. Noether symmetry
d4j d4j dqs þ dpas ¼ 0 vqs vqas
ð8Þ
For nonindependence qs ; pas , it can properly chosen lj , then, the canonical equation of fractional order constrained Hamiltonian systems are obtained by the arbitrariness of the integral interval: 8 v4j vH a > > D q ¼ > < a s vpas þ lj vpas ðs ¼ 1; 2; :::; nÞ > v4j vH a > > : Da pas ¼ lj vqs vqs
ð9Þ
Set ε as an infinitesimal parameter, x0 ; xs ; hs is the generator of infinitesimal transformation, the infinitesimal transformations contain time, generalized coordinates and generalized momenta: t* ¼ t þ εx0 ðt; qs ; pas Þ; q*s ¼ qs þ εxs ðt; qs ; pas Þ; p*as ¼ pas þ εhs ðt; qs ; pas Þ
ð11Þ
In phase space, the total variation of the regular action quantity of fractional systems under the infinitesimal transformations is:
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
Zb DS ¼ a
p*as Da q*s
¼ Iaab DL þ L
H t* ; q*s ; p*as da t*
Zb ½pas Da qs H a ðt; qs ; pas Þda t
a
dDt dt
183
a
ð12Þ
vH a vH a vH a dDt ¼ Iaab pas DDa qs þ Dpas Da qs Dt Dqs Dpas þ ½pas Da qs H a dt vt vpas vpas
Be aware: Dt ¼ εx0 dqs ¼ Dqs q_s Dt ¼ εðxs q_s x0 Þ ¼ εxs dpas ¼ Dpas p_as Dt ¼ εðhs p_as x0 Þ ¼ εhs DDa qs ¼ dDa qs þ Daþ1 qs Dt ¼ Da dqs þ Daþ1 qs Dt ¼ εðDa xs q_s Da x0 Þ ð13Þ Below, we introduce the definition and criterion equations of the Noether symmetric transformation for fractional order constrained Hamiltonian systems (Noether identities). Definition: If the regular action quantity of the fractional order constraint systems is the quasi invariant under the infinitesimal transformation (11) in phase space, that is, for each generator x0 ; xs ; hs, it is always established:
DS ¼ Iaab Da ðDGÞ Iaab Lqs dqs þ Lpas dpas ð14Þ v4j v4j ¼ Iaab Da ðεGÞ Iaab lj dqs þ lj dpas vqs vpas
From the above basic Formula (12) about variational of fractional order constraint Hamiltonian systems, then we can obtain the criterion equation of the Noether quasi symmetric transformation: vH vH pas ðDa xs q_s Da x0 Þ þ ðDa qs Þhs x0 xs vt vqs v4j v4j vH hs þ ðpas Da qs HÞx_ 0 lj xs þ lj h vpas vqs vpas s þ Da G ¼ 0 ð15Þ 5. Noether theorem and inverse theorem 5.1. The Noether theorem of fractional order constrained Hamiltonian systems According to the relation between total variation and isochronous variation, the total variation of the regular action quantity of fractional order singular system (12) can also be expressed as:
_ DS ¼ dS þ SDt vH a vH a ¼ Iaab dqs dpas Da qs dpas þ pas dDa qs þ ðpas Da qs H a ÞDt vqs vpas vH a vH a ab a ¼ Ia Da ½pas þ dqs þ ðpas Da qs H ÞDt þ Da qs dpas þ Da ps dqs vqas vqs
The gauge function G ¼ Gðt; qs ; pas Þ, then, the infinitesimal transformation is the Noether quasi symmetric transformation.
ð16Þ
According to Formula (13), so Formula (16) can be simplification to:
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Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
DS ¼ Da pas xs þ ðpas Da qs H Þx0 þ Da qs vH a vH a hs þ Da ps x vpas vqs s Iaab ε
a
ð17Þ Sum deduced from above, we can easily obtain the Noether theorem for fractional order constrained Hamiltonian systems: Theorem 1: If the generator x0 ðt; qs ; pas Þ; xs ðt; qs ; pas Þ, hs ðt; qs ; pas Þ satisfy the criterion Equation (15) of Noether quasi symmetry for fractional order constraints Hamiltonian systems, then the system have the first integral in the following form: I ¼ pas xs þ ðpas Da qs HÞx0 þ G ¼ const
ð18Þ
5.2. Noether inverse theorem for fractional order constrained Hamiltonian systems It is assumed that a fractional order constrained Hamiltonian systems have the number of l linearly independent conserved quantities in the extended phase space: I s ðt; q; pa Þ ¼ constðs ¼ 1; 2:::; lÞ
ð19Þ
For fractional order constrained Hamiltonian systems, any trajectory is satisfied (9), have:
v4j v4j vH a vH a s Da qs lj h þ D a ps lj vpas vpas s vqs vqs s
xs ¼ 0 ð20Þ The both sides of (19) calculation fractional derivatives about time t, then add it up to (20), have: vI s vI s vI s vH a Da t þ Da qs þ Da pas þ Da qs vt vqs vpas vpas a v4j v4j s vH lj hs þ Da ps lj x vpas s vqs vqs s ¼0
s
s
I ¼ pas xs þ Ls xs0 Gs
ð23Þ
So:
xs0
s s s ¼ L1 s I pas xs þ G
ð24Þ
By (22) and (24) expressions, all xss can be found immediately. Also because: dq*s dqs dqs þ εs xss dq
s ¼ da t* da t da t 1 þ εs da xs0 da t da t s s dxs dx q_s 0 ¼ eð1aÞt εs dt dt
DDa qs ¼
ð25Þ Thus the generator hss ðt; q; pa Þ of pas , it can obtained by xs0 ; xss in Formula (25), such find out xs0 ; xss and hss , as long as they are satisfied: v4j s v4j s x q_s xss þ h p_as xs0 ¼ 0 ð26Þ vqs s vpas s Thus, we obtain the inverse theorem of the regular form Noethor theorem for fractional order constrained Hamiltonian systems: Theorem 2:If the number of l linearly independent of the first integral (motion conserved quantities) (19) about the canonical Equation (10) of holonomic fractional order constraint Hamiltonian systems are known, so, the infinitesimal transformation (11) are formed with xs0 ; xss and hss , it is determined by (24), (22) and (25), as long as Formula (26) is also established, under this transformation, the invariance of the regular action quantity of fractional order constrained Hamiltonian Systems must be suitable for Noether quasi symmetry definitions (14). 6. Illustrated example Example The Lagrange function of fractional order system with two degrees of freedom is: L ¼ ðDa q1 Þq2 ðDa q2 Þq1 þ q21 þ q22
ð21Þ
These relations are always satisfied along arbitrary trajectories, so all coefficients of Da pas should be zero, it is: xs ¼ vI s =vpas
Now take: s
ð22Þ
ð27Þ
0 < a < 1 Please study the Noether symmetry of the system and its inverse problem. The generalized momentum and fractional Hamilton functions of the system are: vL vL pa1 ¼ ¼ q2 ; pa2 ¼ ¼ q1 vDa q1 vDa q2 ð28Þ H a ¼ Pa1 Da q1 þ pa2 Da q2 L ¼ q21 q22
Z. MingLiang / Karbala International Journal of Modern Science 4 (2018) 180e186
The rank of hessian matrix of L is r ¼ 0, Thus, there are two inherent constraints between canonical variables: 41 ¼ pa1 q2 ¼ 0; 42 ¼ pa2 þ q1 ¼ 0
ð29Þ
185
The generator determined by the (38) satisfies the constraint Equation (26), therefore, the Noether quasi symmetry corresponding to fractional order constrained Hamiltonian systems is considered.
The stable consistency conditions are bound by constraints, it can obtained:
7. Conclusions
l1 ¼ q2 ; l2 ¼ q1
In order to solve the large number of singular Lagrange functions problem in modern physics and mechanics, in this paper, Noether symmetry theory has been extended to fractional order constrained Hamiltonian systems based on a fractional factor, A series of new results have been given for the first time, the canonical equation, Noether symmetric criterion equation, conserved quantities and inverse theorem are very concise and completely consistent in form with the conclusion of the classic mechanical systems of integer order. When a is equal to 1, all the results of this paper are degenerated to the Noether theorem of integer order constrained Hamilton systems, it shows that the fractional factor is more extensive. Using the research methods of this paper, The Noether symmetry theory of nonholonomic fractional order constrained Hamiltonian systems and singular fractional electromechanical coupled systems can be further studied.
ð30Þ
The quasi Noether criterion Equation (15) under the definition of quasi symmetry, we have: pas ðDa xs q_s Da x0 Þ þ ðDa qs Þhs þ 2qs xs þ þLx_ 0 þ Da G þ l1 ½h1 x2 þ ðq_2 p_a1 Þx0 þ l2 ½h2 þ x1 ðq_1 þ p_a2 Þx0 ¼ 0
ð31Þ
The upper form follows a set of solutions: x0 ¼ 1; x1 ¼ q2 ; x2 ¼ q1 ; h1 ¼ q1 ; h2 ¼ q2 Gðt; q; pa Þ ¼ 0
ð32Þ
The quasi Noether symmetry leads to the corresponding conserved quantity: I ¼ eð1aÞt pa2 q1 eð1aÞt pa1 q2 þ q21 þ q22 ¼ const ð33Þ Secondly, we study the inverse problem of Noether symmetry. Set up fractional order constrained Hamiltonian systems have integral: I ¼ eð1aÞt pa2 q1 eð1aÞt pa1 q2 þ q21 þ q22 ¼ const ð34Þ By Equations (22) And (24) give the infinitesimal generator of the system are satisfied: x1 q_1 x0 ¼ 0; x2 q_2 x0 ¼ 0 x0 ¼ L1 ½I pa1 ðx1 q_1 x0 Þ pa2 ðx2 q_2 x0 Þ þ G ð35Þ Take Gðt; q; pa Þ ¼ 0, then we get a set of Solutions: x0 ¼ 1; x1 ¼ q2 ; x2 ¼ q1
ð36Þ
According to the expression of regular momentum (25), the other generators are obtained: h1 ¼ x2 ; h2 ¼ x1
ð37Þ
Therefore, the infinitesimal transformation generator of the system can be: x0 ¼ 1; x1 ¼ q2 ; x2 ¼ q1 ; h1 ¼ q1 ; h2 ¼ q2
ð38Þ
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