具有混合导数的分数阶约束Hamilton系统的Noether对称性.pdf

Noether symmetry for fractional constrainedHamiltonian system within mixed derivativesSONG Chuanjing渊School of Mathematical Sciences袁SUST袁Suzhou 215009袁China冤Abstract院 This study investigates the fractional singular system within mixed integer order and Riemann-Liouville frac鄄tional derivatives.The fractional singular Lagrange equation and the fractional constrained Hamilton equation are estab鄄lished.To find the solutions to the differential equations of motion for this singular system袁Noether symmetry method hasbeen studied and the corresponding conserved quantity has been investigated.Namely袁Noether theorems of the fractionalsingular system within mixed integer order and Riemann-Liouville fractional derivatives are established.Key words院 fractional constrained Hamiltonian system曰Noether symmetry曰conserved quantityCLC number院 O316酝砸渊圆园1园冤 杂怎遭躁藻糟贼 悦造葬泽泽蚤枣蚤糟葬贼蚤燥灶院 70H45曰70H33Document code院 粤Article ID院 2096原3289渊圆园23冤园4原园园25原园61IntroductionA system is called a singular system if it is described using a singular Lagrangian.A dynamic system can bedescribed by Lagrangian as well as Hamiltonian.For a singular system袁it is called a constrained Hamiltoniansystem when it is described by Hamiltonian袁because the inherent constraints will exist when it is transformedfrom the Lagrangian to the Hamiltonian through the Legendre transformation1-2.Many important systems in physicsare singular systems or constrained Hamiltonian systems.The problem of seeking conserved quantity of mechanical system is not only of great significance in mathe原matics袁but also reflects the profound physical essence.Symmetry method is an important method to find con原served quantity袁and there are three common symmetry methods院Noether symmetry method袁Lie symmetry methodand Mei symmetry method3-5.Fractional calculus is a hot topic recently.There are four kinds of fractional deriva原tives that are commonly used院Riemann-Liouville fractional derivative袁Caputo fractional derivative袁Riesz-Rie鄄mann-Liouville fractional derivative and Riesz-Caputo fractional derivative 渊see Appendix冤.Many results onfractional variational calculus and fractional Noether symmetry have been obtained6-25.This paper intends to find the conserved quantity of the constrained Hamiltonian system on the basis ofmixed integer and Riemann-Liouville fractional derivatives using Noether symmetry method.Firstly袁the con鄄strained Hamilton equation is established.Then the infinitesimal transformations of the time袁the generalized co鄄ordinates and the generalized momenta are given袁as well as the definitions of Noether symmetry and conservedquantity.And finally the conserved quantity obtained from Noether symmetry is presented.要要要要要要要要要要要要要要要要要要要眼收稿日期演 圆园22原01原12眼基金项目演 国家自然科学基金项目渊12172241曰11972241曰12272248曰11802193冤曰 江苏高校野青蓝工程冶项目曰 江苏省自然科学基金项目渊BK20191454冤眼作者简介演 宋传静渊1987要冤袁女袁河南信阳人袁副教授袁博士袁研究方向院动力学与控制袁E原mail院遥doi院10.12084/j.issn.2096-3289.2023.04.004第 40 卷第 4 期苏 州 科 技 大 学 学 报 渊自 然 科 学 版冤灾燥造援40 晕燥援4圆园23 年 12 月允燥怎则灶葬造 燥枣 杂怎扎澡燥怎 哉灶蚤增藻则泽蚤贼赠 燥枣 杂糟蚤藻灶糟藻 葬灶凿 栽藻糟澡灶燥造燥早赠 渊晕葬贼怎则葬造 杂糟蚤藻灶糟藻 Edition冤Dec援 圆园23圆园23 年苏州科技大学学报渊自然科学版冤2Fractional constrained Hamiltonian systemA dynamic system is described by generalized coordinates qi袁i=1袁2袁噎袁n袁then by finding the extremum ofthe functionalIq渊 窑 冤=t2t1乙L渊t袁q袁q 觶袁t1Dt琢q冤dt渊1冤with the boundary conditions q渊t1冤=q1袁q渊t2冤=q2袁we can get the following fractional Euler-Lagrange equationwithin mixed integer and Riemann-Liouville fractional derivatives鄣L鄣qi-ddt鄣L鄣q 觶i+CtDt2琢鄣L鄣t1Dt琢qi+鄣L渊t2冤鄣t1Dt琢qi窑渊t2-t冤-琢祝渊1-琢冤=0渊2冤where Eq.渊1冤 is called Hamilton action within the mixed integer and Riemann-Liouville fractional derivatives袁q=q1袁q2,噎袁qn袁q1=q11袁q12,噎袁q1n袁q2=q21袁q22,噎袁q2n袁q 觶=q 觶1袁q 觶2,噎袁q 觶n袁t1Dt琢q=t1Dt琢q1袁t1Dt琢q2,噎袁t1Dt琢qn袁q 觶i=dqi/dt袁t1Dt琢qimeans the Riemann-Liouville fractional derivative of qi袁鄣L渊t2冤鄣t1Dt琢qi=鄣L鄣t1Dt琢qi渊t2袁q渊t2冤袁q 觶渊t2冤袁t1Dt琢q渊t2冤冤袁0约琢约1袁qi沂C2渊t1袁t2曰 冤袁L渊 窑 袁 窑 袁 窑 袁 窑 冤沂C2渊t1袁t2伊n伊n伊n曰 冤 and i=1袁2袁噎袁n.Define the generalized momenta and the Hamiltonian aspi=鄣L渊t袁q袁q 觶袁t1Dt琢q冤鄣q 觶i袁pi琢=鄣L渊t袁q袁q 觶袁t1Dt琢q冤鄣t1Dt琢qi渊3冤H=p 窑 q 觶T+p琢窑 渊t1Dt琢q冤T-L渊t袁q袁q 觶袁t1Dt琢q冤渊4冤where p=p1袁p2,噎袁pn袁p琢=p1琢袁p2琢,噎袁pn琢 and i=1袁2,噎袁n.Suppose that all the elements oft1Dt琢q can be solvedand denoted ast1Dt琢qi=xi渊t袁q袁q 觶袁p琢冤袁i=1袁2袁噎袁n袁and only R elements of q 觶 can be solved袁where 0臆R约n.In thiscase袁the Lagrangian L渊t袁q袁q 觶袁t1Dt琢q冤 is called singular.Then by studying Eq.渊3冤袁we can get the inherent con-straint准a=渊t袁q袁p袁p琢冤=0袁a=1袁2袁噎袁n-R袁0臆R约n渊5冤Equation 渊5冤 is called fractional primary constraint.Introducing the Lagrange multiplier 姿a渊t冤袁a=1袁2袁噎袁n-R袁0臆R约n袁from Eqs.渊2冤-渊5冤袁we get the followingfractional constrained Hamilton equation within integer and Riemann-Liouville fractional derivativesq 觶i=鄣H鄣pi+姿a鄣准a鄣pi袁t1Dt琢qi=鄣H鄣pi琢+姿a鄣准a鄣pi琢袁p 觶i=CtDt2琢pi琢+pi琢渊t2冤渊t2-t冤-琢祝渊1-琢冤-鄣H鄣qi-姿a鄣准a鄣qi渊6冤By introducing Poisson bracket袁which is defined asF袁G越鄣F鄣qi鄣G鄣pi-鄣F鄣pi鄣G鄣qi渊7冤where F=F渊t袁q袁p袁p琢冤袁G=G渊t袁q袁p袁p琢冤袁we can get the following consistency condition of the fractional primaryconstraint准a袁H+姿b准a袁准b+鄣准a鄣t+鄣准a鄣pi琢窑 p 觶i琢+鄣准a鄣pi窑CtDt2琢pi琢+pi琢渊t2冤渊t2-t冤-琢祝渊1-琢冤蓸蔀越0渊8冤where a袁b=1袁2袁噎袁n-R袁0臆R约n袁i=1袁2袁噎袁n.Eq.渊8冤 can be used to solve Lagrange multipliers.3Noether symmetry and conserved quantityNoether symmetry means the invariance of the Hamilton action within integer and Riemann-Liouville frac鄄tional derivatives 渊Eq.渊1冤冤 under the infinitesimal transformations.26第 4 期The infinitesimal transformations are given ast 軃 越t+驻t袁q 軈i渊t 軃 冤=qi渊t冤+驻qi袁p 軈i渊t 軃 冤=pi渊t冤+驻pi袁p 軈i琢渊t 軃 冤=pi琢渊t冤+驻pi琢渊9冤and the expanded expressions aret 軃 越t+兹孜0渊t袁q袁p袁p琢冤+o渊兹冤袁q 軈i渊t 軃 冤=qi渊t冤+兹孜i渊t袁q袁p袁p琢冤+o渊兹冤袁p 軈i渊t 軃 冤=pi渊t冤+兹浊i渊t袁q袁p袁p琢冤+o渊兹冤袁p 軈i琢渊t 軃 冤=pi琢渊t冤+兹浊i琢渊t袁q袁p袁p琢冤+o渊兹冤渊10冤where 兹 is a small parameter袁孜0袁孜i袁浊iand 浊i琢are called infinitesimal generators.Let 驻I=I軃-I袁and neglect the higher order of 兹袁we obtain驻I=t 軃2t 軃1乙渊p 軈iq 軈窑i+p 軈i琢t 軃1Dt 軃琢q 軈i-H渊t 軃 袁q 軈 袁p 軈 袁p 軈琢冤冤dt 軃-t2t1乙渊piq 觶i+pi琢t1Dt琢qi-H渊t袁q袁p袁p琢冤冤dt=t2t1乙渊渊pi+驻pi冤渊q 觶i+驻q 觶i冤+渊pi琢+驻pi琢冤渊t1Dt琢qi+t1Dt琢啄qi+驻tddtt1Dt琢qi-1祝渊1-琢冤窑ddt渊渊t-t员冤-琢qi渊t员冤驻t员冤冤-H渊t+驻t袁q+驻q袁p+驻p袁p琢+驻p琢冤冤渊员+ddt驻t冤dt-t2t1乙渊piq 觶i+pi琢t1Dt琢qi-H渊t袁q袁p袁p琢冤冤dt=兹t2t1乙pi孜觶i+pi琢t1Dt琢渊孜i-q 觶i孜0冤+渊pi琢ddtt1Dt琢qi-鄣H鄣t冤孜0-鄣H鄣qi孜i-qi渊t员冤孜0渊t员冤祝渊1-琢冤pi琢ddt渊t-t员冤-琢+渊pi琢t1Dt琢qi-匀冤孜觶园+姿a鄣准a鄣pi琢浊i琢+姿a鄣准a鄣pi浊idt渊11冤where 孜0渊t员冤=孜0渊t员袁q渊t员冤袁p琢渊t员冤袁p渊t员冤冤.The definition of Noether symmetry gives 驻I=0袁i.e.袁pi孜觶i+pi琢t1Dt琢渊孜i-q 觶i孜0冤+渊pi琢ddtt1Dt琢qi-鄣H鄣t冤孜0-鄣H鄣qi孜i+姿a鄣准a鄣pi琢浊i琢+姿a鄣准a鄣pi浊i-qi渊t员冤孜0渊t员冤祝渊1-琢冤pi琢ddt渊t-t员冤-琢+渊pi琢t1Dt琢qi-匀冤孜觶园=0渊12冤Eq.渊12冤 is called Noether identity within mixed integer and Riemann-Liouville fractional derivatives.Furthermore袁if we let 驻I=-t2t1乙ddt渊驻G冤dt袁where 驻G=兹G袁G=G渊t袁q袁p袁p琢冤 is called a gauge function袁thenEq.渊11冤 givespi孜觶i+pi琢t1Dt琢渊孜i-q 觶i孜0冤+渊pi琢ddtt1Dt琢qi-鄣H鄣t冤孜0-鄣H鄣qi孜i+姿a鄣准a鄣pi琢浊i琢+姿a鄣准a鄣pi浊i-qi渊t员冤孜0渊t员冤祝渊1-琢冤pi琢ddt渊t-t员冤-琢+渊pi琢t1Dt琢qi-匀冤孜觶园+G觶=0渊13冤Eq.渊13冤 is called Noether quasi-identity within mixed integer and Riemann-Liouville fractional derivatives.The Noether symmetry leads to a conserved quantity.A quantity C is called a conserved quantity if and onlyif dC/dt=0.Therefore袁we haveTheorem 1For the fractional constrained Hamiltonian system Eq.渊6冤袁if the infinitesimalgenerators 孜0袁孜i袁浊iand 浊i琢satisfy Eq.渊12冤袁then there exists a conserved quantityC=pi孜i+渊pi琢t1Dt琢qi-匀冤孜0+tt1乙pi琢t1D子琢渊孜i-q 觶i孜0冤-渊孜i-q 觶i孜0冤C子Dt2琢pi琢+pi琢渊t2冤 窑渊t2-子冤-琢祝渊1-琢冤d子-qi渊t员冤孜0渊t员冤祝渊1-琢冤tt1乙pi琢dd子渊子-t1冤-琢d子=const渊14冤宋传静院具有混合导数的分数阶约束 Hamilton 系统的 Noether 对称性27圆园23 年苏州科技大学学报渊自然科学版冤ProofUsing Eqs.渊5冤袁渊6冤 and 渊12冤袁we haveddtC=pi孜觶i+p 觶i孜i+渊p 觶i琢t1Dt琢qi+pi琢ddtt1Dt琢qi-鄣H鄣t-鄣H鄣pip 觶i-鄣H鄣qiq 觶i-鄣H鄣pi琢p 觶i琢冤孜0+渊pi琢t1Dt琢qi-匀冤孜觶园+pi琢t1Dt琢渊孜i-q 觶i孜0冤-渊孜i-q 觶i孜0冤CtDt2琢pi琢+pi琢渊t2冤 窑渊t2-t冤-琢祝渊1-琢冤-pi琢祝渊1-琢冤qi渊t员冤孜0渊t员冤ddt渊t-t1冤-琢=-姿a鄣准a鄣pi琢啄pi琢原姿a鄣准a鄣pi啄pi-姿a鄣准a鄣qi啄qi=原姿a啄准a越0This proof is completed.Theorem 2For the fractional constrained Hamiltonian system Eq.渊6冤袁if there exists a gauge function Gsuch that the infinitesimalgenerators 孜0袁孜i袁浊iand 浊i琢satisfy Eq.渊13冤袁then there exists a conserved quantityCG=pi孜i+渊pi琢t1Dt琢qi-匀冤孜0+tt1乙pi琢t1D子琢渊孜i-q 觶i孜0冤-渊孜i-q 觶i孜0冤C子Dt2琢pi琢+pi琢渊t2冤 窑渊t2-子冤-琢祝渊1-琢冤d子-qi渊t员冤孜0渊t员冤祝渊1-琢冤tt1乙pi琢dd子渊子-t1冤-琢d子+G=const渊15冤ProofUsing Eqs.渊5冤袁渊6冤 and 渊13冤袁we can get dCG/dt=0.Remark 1It is obvious that when G=0袁Theorem 2 reduces to Theorem 1.4An exampleThe Lagrangian isL=q 觶1q2-q1q 觶2+渊q1冤2+渊q2冤2+12渊t1Dt琢q1冤2+渊t1Dt琢q2冤2渊16冤try to study its Noether symmetry.Firstly袁we havep1=鄣L鄣q 觶1=q2袁p2=鄣L鄣q 觶2=-q1袁p1琢=鄣L鄣t1Dt琢q1=t1Dt琢q1袁p2琢=鄣L鄣t1Dt琢q2=t1Dt琢q2渊17冤H=p1q 觶1+p2q 觶2+p1琢窑t1Dt琢q1+p2琢窑t1Dt琢q2-L=12渊p1琢冤2+渊p2琢冤2原渊q1冤2原渊q2冤2渊18冤From Eq.渊17冤袁we have准1越p1原q2越0袁准2越p2+q1越0渊19冤Therefore袁the Lagrangian in this example is singular.And we can verify that孜0越原1袁孜1越孜2越0袁浊1越浊2越0袁浊1琢越浊2琢越0袁G=0渊20冤satisfy Eq.渊13冤.Then Theorem 2 givesCG=tt1乙p1琢dd子t1D子琢q1+p2琢dd子t1D子琢q2-q 觶1C子Dt2琢p1琢+p1琢渊t2冤 窑渊t2-子冤-琢祝渊1-琢冤-q 觶2C子Dt2琢p2琢+p2琢渊t2冤 窑渊t2-子冤-琢祝渊1-琢冤d子-12渊p1琢冤2+12渊p2琢冤2+渊q1冤2+渊q2冤2渊21冤28第 4 期5ConclusionBased on mixed integer and Riemann-Liouville fractional derivatives袁the fractional constrained Hamilton e鄄quation is presented firstly.Then the Noether symmetry and the conserved quantity are studied.Theorem 1 andTheorem 2 are new work.AppendixLet f渊t冤 be a function袁t沂t1袁t2袁n is an integer,then the left and the right Riemann-Liouville fractionalderivatives are defined,respectively袁ast1Dt琢f渊t冤=1祝渊n-琢冤ddt蓸蔀ntt1乙渊t-子冤n-琢-1f渊子冤d子袁n-1臆琢约n渊A1冤tDt圆琢f渊t冤=1祝渊n-琢冤-ddt蓸蔀nt圆t乙渊子-t冤n-琢-1f渊子冤d子袁n-1臆琢约n渊A圆冤and the left and the right Caputo fractional derivatives are defined,respectively袁as悦t1Dt琢f渊t冤=1祝渊n-琢冤tt1乙渊t-子冤n-琢-1dd子蓸蔀nf渊子冤d子袁n-1臆琢约n渊A3冤CtDt2琢f渊t冤=1祝渊n-琢冤t圆t乙渊子-t冤n-琢-1-dd子蓸蔀nf渊子冤d子袁n-1臆琢约n渊A4冤Specially袁when 琢寅1袁we havet1Dt员f渊t冤越悦t1Dt员f渊t冤越df渊t冤/dt=f觶渊t冤袁tDt圆员f渊t冤=CtDt2员f渊t冤越-df渊t冤/dt=-f觶渊t冤渊A5冤References院1 LI Z P.Contrained Hamiltonian Systems and Their Symmetrical PropertiesM.Beijing院Beijing Polytechnic University Press袁1999.渊in Chinese冤2 LI Z P袁JIANG J H.Symmetries in Constrained Canonical SystemsM.Beijing院Science Press袁2002.渊in Chinese冤3 MEI F X.Analytical Mechanics 渊II冤M.Beijing院Beijing Institute of Technology Press袁2013.4 MEI F X袁WU H B.Dynamics of Constrained Mechanical SystemsM.Beijing院Beijing Institute of Technology Press袁2009.5 MEI F X袁WU H B袁ZHANG Y F.Symmetries and conserved quantities of constrained mechanical systemsJ.International Journal of Dynamicsand Control袁2014袁2院285-303.6 RIEWE F.Nonconservative Lagrangian and Hamiltonian mechanicsJ.Physical Review E袁1996袁53院1890原1899.7 KLIMEK M.Lagrangian and Hamiltonian fractional sequential mechanicsJ.Czechoslovak Journal of Physics袁2002袁52院1247原1253.8 AGRAWAL O P.Formulation of Euler原Lagrange equations for fractional variational problemsJ.Journal of Mathematical Analysis and Applica原tions袁2002袁272院368原379.9 AGRAWAL O P.Generalized variational problems and Euler-Lagrange equationsJ.Computers and Mathematics with Applications袁2010袁59院1852-1864.10 MUSLIH S I袁BALEANU D.Hamiltonian formulation of systems with linear velocities within Riemann原Liouville fractional derivativesJ.Journalof Mathematical Analysis and Applications袁2005袁304院599原606.11 RABEI E M袁NAWAFLEH K I袁HIJJAWI R S袁et al.The Hamilton formalism with fractional derivativesJ.Journal of Mathematical Analysis andApplications袁2007袁327院891-897.12 LUO S K袁XU Y L.Fractional Birkhoffian mechanicsJ.Acta Mechanica袁2015袁226院829-844.13 ZHANG H B袁CHEN H B.Generalized variational problems and Birkhoff equationsJ.Nonlinear Dynamics袁2016袁83院347-354.14 SONG C J袁ZHANG Y.Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applicationsJ.Fractional Calculusand Applied Analysis袁2018袁21院509-526.15 DING J J袁ZHANG Y.Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz typeJ.Chinese Physics B袁2020袁29院044501.16 ZHOU Y袁ZHANG Y.Noether symmetries for fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivativesJ.宋传静院具有混合导数的分数阶约束 Hamilton 系统的 Noether 对称性29圆园23 年苏州科技大学学报渊自然科学版冤Acta Mechanica袁2020袁231院3017-3029.17 SONG C J袁CHENG Y.Noether symmetry method for Hamiltonian mechanics involving generalized operatorsJ.Advances in Mathematical袁Physics袁2021袁2021院1959643.18 DING J J袁ZHANG Y.Noether爷s theorem for fractional Birkhoffian system of Herglotz type with time delayJ.Chaos袁Solitions and Fractals袁2020袁138院109913.19 SONG C J袁ZHANG Y.Perturbation to Noether symmetry for fractional dynamical systems of variable orderJ.Indian Journal of Physics袁2019袁93院1057-1067.20 TIAN X袁HANG Y.Noether爷s theorem for fractional Herglotzvariational principle in phase spaceJ.Chaos袁Solitions and Fractals袁2019袁119院50-54.21 SONG C J袁ZHANG Y.Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systemsJ.International Journal ofNon-Linear Mechanics袁2017袁90院32-38.22 YANG B袁ZHANG Y.Noether爷s theorem for fractional Birkhoffian systems of variable orderJ.Acta Mechanica袁2016袁227院2439-2449.23 ZHAI X H袁ZHANG Y.Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delayJ.Communications inNonlinear Scienceand Numerical Simulation袁2016袁36院81-97.24 DING J F袁ZHANG Y.Noether symmetrical perturbation and adiabatic invariants for systems of generalized classical mechanics based on El-Nabulsi爷s fractional modelJ.Journal of Suzhou University of Science and Technology 渊Natural Science Edition冤袁2018袁35渊4冤院9-17.渊in Chinese冤25 CHEN J袁ZHANG Y.Noether symmetries and perturbation for nonholonomic systems based on El-Nabulsi fractional order modelsJ.Journal ofSuzhou University of Science and Technology 渊Natural Science Edition冤袁2015袁32渊1冤院8-11.渊in Chinese冤具有混合导数的分数阶约束 Hamilton 系统的 Noether 对称性宋传静渊苏州科技大学 数学科学学院袁江苏 苏州 215009冤摘要院 研究了混合整数阶和 Riemann-Liouville 分数阶导数下的分数阶奇异系统遥 建立了分数阶奇异Lagrange 方程和分数阶约束 Hamilton 方程遥 为了寻找该奇异系统微分方程的解袁 论文研究了 Noether 对称性袁并得到了相应的守恒量遥 即袁建立了混合整数阶和 Riemann-Liouville 分数阶导数下的分数阶奇异系统的Noether 定理遥关键词院 分数阶约束 Hamilton 系统曰Noether 对称性曰守恒量责任编辑院谢金春30