Lagrangians and Hamiltonians for one-dimensional systems

汉密尔顿原理The Hamiltonian principle, also known as Hamilton's principle, is a fundamental principle in classical mechanics. It states that the dynamics of a physical system are determined by a single function, known as the Hamiltonian. This principle was formulated by Sir William Rowan Hamilton in 1834 and is a powerful tool for understanding the behavior of a wide range of physical systems.汉密尔顿原理，也称为汉密尔顿原则，是古典力学中的基本原理。

它表明物理系统的动力学是由一个称为汉密尔顿量的单个函数所决定的。

这一原理是由威廉·罗恩·汉密尔顿爵士于1834年提出的，是理解各种物理系统行为的有力工具。

One of the key insights of the Hamiltonian principle is that it provides a more general formulation of the laws of motion than the standard Newtonian approach. While Newton's laws are suitable for describing the motion of simple, low-energy systems, the Hamiltonian approach can be applied to more complex systems, including those involving relativistic effects and quantum mechanics.汉密尔顿原理的一个关键见解是，它提供了比标准牛顿方法更一般的运动定律公式。

昆明“PEP”2024年小学三年级上册英语下册试卷(含答案)考试时间：80分钟（总分:110）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题共100分)1. 填空题：The ________ was a major event in the history of international relations.2. 选择题：What is the capital of Belgium?A. BrusselsB. AntwerpC. GhentD. Bruges答案: A3. 填空题：The first human-made object to reach the moon was _______. (探测器)4. 听力题：The ice is ______ in the glass. (cold)5. 选择题：What is the hardest natural substance on Earth?A. GoldB. IronC. DiamondD. Silver答案:C6. 填空题：Certain plants can ______ (提供) essential oils.7. 填空题：I can ______ (逛) the mall with my friends.The __________ is a crucial area for studying biodiversity.9. 听力题：The _______ can be a great topic for science projects.10. 选择题：What is the main ingredient of the sun?A. OxygenB. HydrogenC. HeliumD. Carbon11. 选择题：What is the name of the plant that grows in water?A. CactusB. LilyC. OakD. Pine答案: B12. 选择题：What is the primary ingredient in a cake?A. FlourB. SugarC. EggsD. Butter13. 选择题：What is the name of the large ocean between Africa and Australia?A. Atlantic OceanB. Indian OceanC. Arctic OceanD. Pacific Ocean14. 听力题：The _______ of a liquid can change when heated.15. 选择题：What is the capital of Cambodia?a. Phnom Penhb. Siem Reapc. Battambangd. Sihanoukville答案:aMy aunt, ______ (我的阿姨), travels around the world.17. 填空题：I have a pet _______ that loves to run (我有一只喜欢跑的宠物_______).18. 填空题：I received a ________ (生日派对) invitation with a cute ________ (设计).19. 填空题：The ______ (金鱼) swims in circles in its bowl.20. 选择题：How many letters are in the word "alphabet"?A. 6B. 7C. 8D. 9答案:C21. 填空题：We have a ______ (丰富的) resource center at school.22. 填空题：A wolf hunts in _______ (群体).23. 填空题：My _____ (小鸟) sings sweetly in the morning.24. 选择题：What do you call a young chinchilla?A. KitB. PupC. CubD. Calf25. 选择题：Which animal is known for its long neck?A. ElephantB. GiraffeC. LionD. Tiger答案:BMetamorphic rocks are formed from existing rocks under ______ and pressure.27. 选择题：What is the name of the famous American singer known for "Shake It Off"?A. Taylor SwiftB. Katy PerryC. Miley CyrusD. Demi Lovato答案：A28. 填空题：I can ______ (进行) experiments in the lab.29. 填空题：My ________ (玩具) is a gift from my grandmother.30. 选择题：What is the name of the famous artist known for his abstract paintings?A. PicassoB. Van GoghC. Da VinciD. Monet答案:A31. 填空题：The squirrel gathers nuts before _________. (冬天)32. 听力题：I can ________ (jump) very high.33. 填空题：The ______ (生态挑战) require innovative solutions.34. 填空题：Every evening, I take a walk in the ______ (社区). I enjoy seeing my neighbors and saying ______ (你好).35. 听力题：The chemical symbol for lithium is ______.36. 听力题：In a chemical reaction, reactants are transformed into ______.The ____ is known for its ability to glide through the air.38. 填空题：A _____ (植物行为) can reveal adaptation strategies.39. 选择题：What do you call the story of someone's life?A. BiographyB. NovelC. FictionD. Poem40. 填空题：I enjoy _______ (与朋友一起)外出.41. 听力题：The chemical symbol for cesium is __________.42. 听力题：I have a _____ (篮球) for practice.43. 填空题：I enjoy ________ (看书) before bedtime.44. 填空题：I enjoy drawing pictures of _______ (我喜欢画_______的图画).45. 填空题：I love to watch videos about _________ (玩具) on the internet.46. 填空题：The _____ (绿色能源) movement promotes plant-based solutions.47. 听力题：A ________ is a large area of land with a specific climate.48. 选择题：What do you call a person who studies the weather?A. MeteorologistB. ClimatologistC. GeologistD. Biologist答案:AA molecule is formed when two or more _____ (atoms) bond together.50. 听力题：The man has a funny ________.51. 听力题：The _______ can thrive in challenging conditions.52. 选择题：What household item is used to clean floors?A. Vacuum CleanerB. RefrigeratorC. OvenD. Microwave答案: A53. 听力题：I have a ___ (pencil) in my bag.54. 听力题：The freezing point of water is ______ degrees Fahrenheit.55. 选择题：What part of the plant absorbs water?A. LeafB. StemC. RootD. Flower答案: C56. 填空题：The ______ (猫) sleeps a lot during the day.57. 填空题：She is a dancer, ______ (她是一位舞者), and performs on stage.58. 选择题：What is the name of the musical instrument with keys?A. GuitarB. FluteC. PianoD. Drums答案: CAn octopus has ______ (触手) and lives in the sea.60. 填空题：My mom is a great __________ (家长) who cares deeply for us.61. 选择题：What do we celebrate on New Year's Eve?A. BirthdaysB. New YearC. ChristmasD. Halloween答案: B62. 填空题：The __________ (道路) is busy during rush hour.63. 选择题：What do we use to brush our teeth?A. ShampooB. ToothbrushC. SoapD. Comb答案:B64. 听力题：The chemical symbol for gallium is _______.65. 听力题：A reaction that absorbs heat is ______.66. 填空题：He is a _____ (发明家) aiming for sustainability solutions.67. 听力题：The dog likes to _____ (bark/sleep).68. 填空题：The _______ (海马) is a unique fish that swims upright.69. 选择题：What is the name of the ocean on the west coast of the United States?A. Atlantic OceanB. Indian OceanC. Arctic OceanD. Pacific Ocean答案: D70. 选择题：What is the name of the famous music festival held in the summer?A. CoachellaB. WoodstockC. LollapaloozaD. Bonnaroo71. 选择题：What do we call a young seal?A. PupB. CalfC. KitD. Chick答案:A. Pup72. 听力题：A solid has a _____ shape and volume.73. 填空题：The __________ (历史的探索) reveals insights.74. 填空题：The _______ (小鹦鹉) can mimic human speech.75. 填空题：The _____ (花坛) is filled with roses.76. 填空题：I planted a _____ (树) in my backyard.77. 填空题：Many plants have important ______ (生态功能).78. 选择题：What do we call the feeling of being afraid?A. JoyB. AngerC. FearD. Sadness答案: C. FearA ______ is a representation of scientific knowledge.80. 填空题：The __________ (两次世界大战) changed global politics.81. 听力题：The rainbow is _____ in the sky. (bright)82. 填空题：My toy ______ is a bright yellow duck.83. 选择题：What do we call a story that is not true?A. FactB. FictionC. TruthD. Reality答案:B84. 听力题：My favorite color is _____ (blue/sit).85. 选择题：What is the name of the bird that cannot fly and is native to New Zealand?A. OstrichB. KiwiC. EmuD. Cassowary答案：B86. 听力题：A solution that contains a large amount of solute is said to be _______.87. 填空题：The _____ (开花) of the cherry blossom tree is celebrated.88. 选择题：What is the name of the highest waterfall in the world?A. Angel FallsB. Niagara FallsC. Victoria FallsD. Yosemite Falls答案: AThe __________ (历史的传承) enriches our culture.90. 听力题：I want to _____ (ride) a horse.91. 选择题：What do we call the layer of the Earth that is made up of molten rock?A. CrustB. MantleC. CoreD. Lithosphere92. 选择题：What is the main gas that plants use for photosynthesis?A. OxygenB. Carbon dioxideC. NitrogenD. Helium答案:B. Carbon dioxide93. 听力题：A ______ is a type of energy associated with moving objects.94. 填空题：The _____ (zucchini) is a popular vegetable.95. 选择题：What is the term for a baby chicken?A. PigletB. CalfC. ChickD. Lamb答案:C96. 听力题：The chemical symbol for platinum is ______.97. 听力题：A ______ is a systematic investigation of a phenomenon.98. 填空题：The ________ is a very special plant.99. 听力题：The first successful flight of a hot air balloon was in _______.100. 听力题：The chemical formula for cobalt(II) nitrate is _____.。

a r X i v :h e p -t h /9705178v 1 23 M a y 1997Generalized Chiral QED 2:Anomaly and Exotic StatisticsFuad M.SaradzhevInstitute of Physics,Academy of Sciences of Azerbaijan,Huseyn Javid pr.33,370143Baku,AZERBAIJANABSTRACTWe study the influence of the anomaly on the physical quantum picture of the generalized chiral Schwinger model defined on S 1.We show that the anomaly i)results in the background linearly rising electric field and ii)makes the spectrum of the physical Hamiltonian nonrelativistic without a massive boson.The physical matter fields acquire exotic statistics.We construct explicitly the algebra of the Poincare generators and show that it differs from the Poincare one.We exhibit the role of the vacuum Berry phase in the failure of the Poincare algebra to close.We prove that,in spite of the background electric field,such phenomenon as the total screening of external charges characteristic for the standard Schwinger model takes place in the generalized chiral Schwinger model,too.PACS numbers:03.70+k ,11.10.Mn.1IntroductionThe two-dimensional QED with massless fermions,i.e.the Schwinger model(SM),demonstrates such phenomena as the dynamical mass generation and the total screening of the charge[1].Although the Lagrangian of the SM contains only masslessfields,a massive bosonfield emerges out of the interplay of the dynamics that govern the originalfields.This mass generation is due to the complete compensation of any external charge inserted into the vacuum.In the chiral Schwinger model(CSM)[2,3]the right and left chiral components of the fermionic field have different charges.The left-right asymmetric matter content leads to an anomaly.At the quantum level,the local gauge symmetry is not realized by a unitary action of the gauge symmetry group on Hilbert space.The Hilbert space furnishes a projective representation of the symmetry group[4,5,6].In this paper,we aim to study the influence of the anomaly on the physical quantum picture of the CSM.Do the dynamical mass generation and the total screening of charges take place also in the CSM?Are there any new physical effects caused just by the left-right asymmetry?These are the questions which we want to answer.To get the physical quantum picture of the CSM we needfirst to construct a self-consistent quantum theory of the model and then solve all the quantum constraints.In the quantization procedure,the anomaly manifests itself through a special Schwinger term in the commutator algebra of the Gauss law generators.This term changes the nature of the Gauss law constraint:instead of beingfirst-class constraint,it turns into second-class one.As a consequence,the physical quantum states cannot be defined as annihilated by the Gauss law generator.There are different approaches to overcome this problem and to consistently quantize the CSM. The fact that the second class constraint appears only after quantization means that the number of degrees of freedom of the quantum theory is larger than that of the classical theory.To keep the Gauss law constraintfirst-class,Faddeev and Shatashvili proposed adding an auxiliaryfield in such a way that the dynamical content of the model does not change[7].At the same time,after quantization it is the auxiliaryfield that furnishes the additional”irrelevant”quantum degrees of freedom.The auxiliaryfield is described by the Wess-Zumino term.When this term is added to the Lagrangian of the original model,a new,anomaly-free model is obtained.Subsequent canonical quantization of the new model is achieved by the Dirac procedure.For the CSM,the correspondig WZ-term is not defined uniquely.It contains the so called Jackiw-Rajaraman parameter a>1.This parameter reflects an ambiguity in the bosonization procedure and in the construction of the WZ-term.The spectrum of the new,anomaly-free model turns out to be relativistic and contains a relativistic boson.However,the mass of the boson also depends on the Jackiw-Rajaraman parameter[2,3].This mass corresponds therefore to the”irrelevant”quantum degrees of freedom.The quantum theory with such a parameter in the spectrum is not physical,i.e. thatfinal version of the quantum theory which we would like to get.The latter should not contain any nonphysical parameters,otherwise one can not say anything about a physical quantum picture.In another approach also formulated by Faddeev[8],the auxiliaryfield is not added,so the quantum Gauss law constraint remains second-class.The standard Gauss law is assumed to be regained as a statement valid in matrix elements between some states of the total Hilbert space,and it is the states that are called physical.The theory is regularized in such a way that the quantum Hamiltonian commutes with the nonmodified,i.e.second-class quantum Gauss law constraint.The spectrum turns out to be non-relativistic[9,10].Here,we follow the approach given in our previous work[11].The pecularity of the CSM is that its anomalous behaviour is trivial in the sense that the second class constraint which appears afterquantization can be turned intofirst class by a simple redefinition of the canonical variables.This allows us to formulate a modified Gauss law to constrain physical states.The physical states are gauge-invariant up to a phase,the phase being1-cocycle of the gauge symmetry group algebra.In [12,13,14],the modification of the Gauss law constraint is obtained by making use of the adiabatic approach.Contrary to[11]where the CSM is defined on R1,we suppose here that space is a circle of lengthL,−L2,so space-time manifold is a cylinder S1×R1.The gaugefield then acquires aglobal physical degree of freedom represented by the non-integrable phase of the Wilson integral on S1.We show that this brings in the physical quantum picture new features of principle.Another way of making two-dimensional gaugefield dynamics nontrivial is byfixing the spatial asymptotics of the gaugefield[15,16].If we assume that the gaugefield defined on R1diminishes rather rapidly at spatial infinities,then it again acquires a global physical degree of freedom.We will see that the physical quantum picture for the model defined on S1is equivalent to that obtained in[15,16].We consider the general version of the CSM with a U(1)gaugefield coupled with different charges to both chiral components of a fermionicfield.We show that the charges are not arbitrary,but satisfy a quantization condition.The SM where these charges are equal is a special case of the generalized CSM.This will allow us at each step of our consideration to see the distinction between the two models.We work in the temporal gauge A0=0in the framework of the canonical quantization scheme and the Dirac’s quantization method for the constrained systems[17].We use the system of units where c=1.In Section2,we quantize our model in two steps.First,the matterfields are quantized, while A1is handled as a classical backgroundfield.The gaugefield A1is quantized afterwords,using the functional Schrodinger representation.We derive the anomalous commutators with nonvanishing Schwinger terms which indicate that our model is anomalous.In Section3,we show that the Schwinger term in the commutator of the Gauss law generators is removed by a redefinition of these generators and formulate the modified quantum Gauss law constraint.We prove that this constraint can be also obtained by using the adiabatic approximation and the notion of quantum holonomy.In Section4,we construct the physical quantum Hamiltonian consistent with the modified quan-tum Gauss law constraint,i.e.invariant under the modified gauge transformations both topologically trivial and non-trivial.We introduce the modified topologically non-trivial gauge transformation op-erator and defineθ–states which are its eigenstates.We consider in detail the case of the SM and demonstrate its equivalence to the freefield theory of a massive scalarfield.For the generalized CSM,we define the exotic statistics matterfield and reformulate the quantum theory in terms of thisfield.In Section5,we construct two other Poincare generators,i.e.the momentum and the boost.We act in the same way as before with the Hamiltonian,namely we define the physical generators as those which are invariant under both topologically trivial and non-trivial gauge transformations.We show that the algebra of the constructed generators is not a Poincare one and that the failure of the Poincare algebra to close is connected to the nonvanishing vacuum Berry curvature.In Section6,we study the charge screening.We introduce external charges and calculate(i)the energy of the ground state of the physical Hamiltonian with the external charges and(ii)the current density induced by these charges.Section7contains our conclusions and discussion.2Quantization Procedure2.1Classical TheoryThe Lagrangian density of the generalized CSM isL=−10,1,γ0=σ1,γ1=−iσ2,γ0γ1=γ5=σ3,σi(i=2(1±γ5)ψ.In the temporal gauge A0=0,the Hamiltonian density isH=H EM+H F,(2) where H EM=12)=A1(L2)=ψ±(L¯he±λ}ψ±,generated byG=∂1E+e+j++e−j−,λbeing a gauge function,as well as under global gauge transformations of the right-handed and left-handed Diracfields which are generated byQ±=e± L/2−L/2dxj±(x).Due to the gauge invariance,the Hamiltonian density is not uniquely determined.On the con-strained submanifold G≈0of the full phase space,the Hamiltonian density˜H=H+v H·G,(4) where v H is an arbitrary Lagrange multiplier which can be any function of thefield variables and their momenta,reduces to the Hamiltonian density H.In this sense,our theory cannot distinguish between H and˜H,and so both Hamiltonian densities are physically equivalent to each other.For arbitrary e+,e−the gauge transformations do not respect the boundary conditions 3.The gauge transformations compatible with the boundary conditions must be either of the formλ(L2)+¯h2πe+=N,N∈Z,(6)or of the formλ(L2)+¯h2πe−=N∈Z.(7)Eqs.6or7imply the charge quantization condition for our system.Without loss of generality, we choose the condition 6.For N=1,e−=e+and we have the standard Schwinger model.For N=0,we get the model in which only the right-handed component of the Diracfield is coupled to the gaugefield.From Eq.5we see that the gauge transformations under consideration are divided into topo-logical classes characterized by the integer n.Ifλ(L2),then the gauge transformation istopologically trivial and belongs to the n=0class.If n=0it is nontrivial and has winding number n.Given Eq.5,the nonintegrable phaseΓ(A)=exp{i¯he+L b(t)}.In contrast toΓ(A),the line integralb(t)=1e+Ln.By a non-trivial gauge transformation of the formg n=exp{i2πe+L].The configurations b=0and b=¯h2πe+L.2.2Quantization and AnomalyThe eigenfunctions and the eigenvalues of thefirst quantized fermionic Hamiltonians ared± x|n;± =±εn,± x|n;± ,wherex|n;± =1L exp{i¯hεn,±·x},εn,±=2π2π).We see that the spectrum of the eigenvalues depends on b.For e+b Le+L,the energies ofεn,+decrease by¯h2πL N.Some of energy levels change sign.However,the spectrum atthe configurations b=0and b=¯h2π2π¯h (and e−b L2π¯h]and{e±b L2π¯h],a†n|vac;A;+ =0for n≤[e+b Landb n |vac;A ;− =0for n ≤[e −b L2π¯h ].(11)Excited states are constructed by operating creation operators on the Fock vacuum.In the ζ–function regularization scheme,we define the action of the functional derivative on first quantized fermionic kets and bras byδδA 1(x )|n ;± ·|λεm,±|−s/2,n ;±|←δδA 1(x )|m ;± m ;±|·|λεm,±|−s/2.From 8we get the action ofδδA 1(x )a n =−lim s →0m ∈Zn ;+|δδA 1(x )a †n=lims →0m ∈Zm ;+|δδA 1(x )on b n ,b †n can be written analogously.Next we define the quantum fermionic currents and fermionic parts of the second-quantized Hamiltonian asˆj s ±(x )=12L /2−L /2dx (ψ†s ±d ±ψs ±−ψs ±d ⋆±ψ†s±).Substituting 8into these expressions,we obtainˆj s ±(x )=n ∈Z1Lnx }ρs ±(n ),whereρs +(n )≡k ∈Z12[b †k ,b k +n ]−·|λεk,−|−s/2|λεk +n,−|−s/2are momentum space charge density (or current)operators,andˆH s ±(x )=n ∈Z1Lnx }H s±(n ),H s ±(n )≡H s 0,±(n )∓e ±bρs±(n ),(12)whereH s0,+(n)≡¯hπ2[a†k,a k+n]−·|λεk,+|−s/2|λεk+n,+|−s/2,H s0,−(n)≡¯hπ2[b k+n,b†k]−·|λεk,−|−s/2|λεk+n,−|−s/2. The charges corresponding to the currentsˆj s±(x)areˆQ s±=e± L/2−L/2dxˆj s±(x)=e±ρs±(0).With Eqs.10and11,we have for the vacuum expectation values:vac;A;±|ˆj±(x)|vac;A;± =−12(ξ++ξ−),whereη±≡±lim s→01λ k∈Z|λεk,±|−s+1.Taking the sums,we obtainη±=±22π¯h}−1L(({e±b L2)2−12η±,ˆQ ±=e±:ρ±(0):−L2ξ±,where double dots indicate normal ordering with respect to|vac,A ,ˆH 0,+=¯h2π2π¯h]ka†k a k|λεk,+|−s− k≤[e+b LLlims→0{k>[e−b L2π¯h]kb†k b k|λεk,−|−s}and:ρ+(0):=lims→0{ k>[e+b L2π¯h]a k a†k|λεk,+|−s},:ρ−(0):=lims→0{ k≤[e−b L2π¯h]b k b†k|λεk,−|−s}.The operators:ˆj±(x):and:ˆH±:are well defined when acting onfinitely excited states which have only afinite number of excitations relative to the Fock vacuum.To construct the quantum electromagnetic Hamiltonian,we quantize the gaugefield using the functional Schrodinger representation.In this representation,when the vacuum and excited fermionic Fock states are functionals of A1,the gaugefield operators are represented asˆA1(x)→A1(x),ˆE(x)→−i¯hδL pxαp.Since A1(x)is a real function,αp satisfiesαp=α⋆−p.The Fourier expansion for the canonical momentum conjugate to A1(x)is thenˆE(x)=1L¯h p∈Z p=0e−i2πdαp, whereˆπb≡−i¯h dL exp{i2πL ¯h2ddb−1dα−p+qd2Lˆπ2b−1dαqd2(ξ++ξ−).If we multiply two operators that arefinite linear combinations of the fermionic creation and annihilation operators,theζ–function regulated operator product agrees with the naive product. However,if the operators involve infinite summations their naive product is not generally well defined. We then define the operator product by mutiplying the regulated operators with s large and positive and analytically continue the result to s=0.In this way we obtain the following relations[ρ±(m),ρ±(n)]−=±mδm,−n,(15) [H0,±(n),H0,±(m)]−=±¯h2π[ˆH0,±,ρ±(m)]−=∓¯h2πdbρ±(m)=0,d2π¯hδp,±m,d2π¯hδp,±m,(p>0).(16) The quantum Gauss operator isˆG=ˆG0+2πLpx−ˆG−(p)e−i2πLe+ρN(0),ˆG ±(p)≡¯h pd2πρN(±p)andρN=ρ++Nρ−is momentum space total charge density operator.Using15and16,we easily get thatρ+(±p)(andρ−(±p))are gauge invariant.For example, forρ+(±p)we have:[ˆG+(p),ρ+(±q)]−=0,[ˆG−(p),ρ+(±q)]−=0,(p>0,q>0).The operatorsˆG±(p)don’t commute with themselves,[ˆG+(p),ˆG−(q)]−=(1−N2)e2+L24π2d3Quantum Constraints3.1Quantum SymmetryIn non-anomalous gauge theories,Gauss law is considered to be valid for physical states only.This identifies physical states as those which are gauge-invariant.The problem with the anomalous be-haviour of the generalized CSM,in terms of states in Hilbert space,is apparent:owing to theSchwinger terms we cannot require that states be annihilated by the Gauss law generators ˆG±(p ).Let us represent the action of the topologically trivial gauge transformations by the operatorsU 0(τ)=exp {i¯hp>0(ˆG+τ++ˆG −τ−)}(17)with τ0,τ±(p )smooth,thenU −10(τ)α±p U 0(τ)=α±−ipτ∓(p ),U −1(τ)d dα±p∓i 2π)2τ±(p ),(p >0).The composition law for the operators U 0isU 0(τ(1))U 0(τ(2))=exp {2πiω2(τ(1),τ(2))}U 0(τ(1)+τ(2)),whereω2(τ(1),τ(2))≡−i2π¯h )2p>0p (τ(1)−τ(2)+−τ(1)+τ(2)−)is a 2-cocycle of the gauge group algebra.Thus for N =±1we are dealing with a projectiverepresentation.The 2-cocycle ω2(τ(1),τ(2))is trivial,since it can be removed by a simple redefinition of U 0(τ).Indeed,the modified operators˜U0(τ)=exp {i 2πα1(γ;τ)}·U 0(τ),(18)whereα1(γ,τ)≡−12π¯h )2p>0(α−p τ−−αp τ+)is a 1-cocycle,satisfy the ordinary composition law˜U0(τ(1))˜U 0(τ(2))=˜U 0(τ(1)+τ(2)),i.e.the action of the topologically trivial gauge transformations represented by 18is unitary.The modified Gauss law generators corresponding to 18areˆ˜G±(p )=ˆG ±(p )±18π2α±p .(19)The generators ˆ˜G±(p )commute:[ˆ˜G+(p ),ˆ˜G −(q )]−=0.This means that Gauss law can be maintained at the quantum level for N=±1,too.We define physical states as those which are annihilated byˆ˜G±(p)[11]:ˆ˜G(p)|phys;A =0.(20)±The zero componentˆG0is a sum of quantum generators of the global gauge transformations of the right-handed and left-handed fermionicfields,so the other quantum constraints are:ρ±(0):|phys;A =0.(21) It follows from20that the physical states|phys;A respond to a gauge transformation from the zero topological class with a phase:U0(τ)|phys;A =exp{−i2πα1(γ;τ)}|phys;A .(22) Only for models without anomaly,i.e.for N=±1,this equation translates into the statement that physical states are gauge invariant.Equation22expresses in an exact form the nature of anomaly in the CSM.At the quantum level the gauge invariance is not broken,but realized projectively.The1-cocycleα1occuring in the projective representation contributes to the commutator of the Gauss law generators by a Schwinger term and produces therefore the anomaly.3.2Adiabatic ApproachLet us show now that we can come to the quantum constraints20and21in a different way,using the adiabatic approximation[23,24].In the adiabatic approach,the dynamical variables are divided into two sets,one which we call fast variables and the other which we call slow variables.In our case, we treat the fermions as fast variables and the gaugefields as slow variables.Let A1be a manifold of all static gaugefield configurations A1(x).On A1a time-dependent gaugefield A1(x,t)corresponds to a path and a periodic gaugefield to a closed loop.We consider the fermionic part of the second-quantized Hamiltonian:ˆH F:which depends on t through the background gaugefield A1and so changes very slowly with time.We consider next the periodic gaugefield A1(x,t)(0≤t<T).After a time T the periodicfield A1(x,t)returns to its original value:A1(x,0)=A1(x,T),so that:ˆH F:(0)=:ˆH F:(T).At each instant t we define eigenstates for:ˆH F:(t)by:ˆH F:(t)|F,A(t) =εF(t)|F,A(t) .The state|F=0,A(t) ≡|vac,A(t) is a ground state of:ˆH F:(t),:ˆH F:(t)|vac,A(t) =0.The Fock states|F,A(t) depend on t only through their implicit dependence on A1.They are assumed to be periodic in time,|F,A(T) =|F,A(0) ,orthonormalized,F′,A(t)|F,A(t) =δF,F′,and nondegenerate.The time evolution of the wave function of our system(fermions in a background gaugefield)is clearly governed by the Schrodinger equation:∂ψ(t)i¯h¯h T0dt·εF(t),whileT0dt L/2−L/2dx˙A1(x,t) F,A(t)|iδγBerryF≡δA1(x,t)|F,A(t) ,(24) then= T0dt L/2−L/2dx˙A1(x,t)A F(x,t).γBerryFWe see that upon parallel transport around a closed loop on A1the Fock state|F,A(t) acquiresan additional phase which is integrated exponential of A F(x,t).Whereas the dynamical phaseγdynF provides information about the duration of the evolution,the Berry’s phase reflects the nontrivial holonomy of the Fock states on A1.However,a direct computation of the diagonal matrix elements ofδδδA1(x,t)A F(y,t)−2π2¯h2 n>01L n(x−y))=(1−N2)e2+2ǫ(x−y)−1The corresponding U(1)connection is easily deduced asA F=0(x,t)=−12 T0dt L/2−L/2dx L/2−L/2dy˙A1(x,t)F F=0(x,y,t)A1(y,t).In terms of the Fourier components,the connection A F=0is rewritten as vac,A(t)|ddα±p(t)|vac,A(t) ≡A±(p,t)=±(1−N2)e2+L2pα∓p,so the nonvanishing curvature isF+−(p)≡d dαpA−=(1−N2)e2+L2p.A parallel transportation of the vacuum|vac,A(t) around a closed loop in(αp,α−p)–space(p>0) yields back the same vacuum state multiplied by the phaseγBerry F=0=(1−N2)e2+L2piαp˙α−p.This phase is associated with the projective representation of the gauge group.For N=±1,when the representation is unitary,the curvature F+−and the Berry phase vanish.As mentioned in the beginning of this Section,the projective representation is trivial and the2-cocycle in the composition law of the gauge transformation operators can be removed by a redefinition of these operators.Analogously,if we redefine the momentum operators asddα±p≡d8π2¯h21 dα±p|vac,A(t) =0,˜F+−=˜ddαp˜A−=0.However,the nonvanishing curvature F+−(p)shows itself in the algebra of the modified momentum operators which are noncommuting:[˜ddα−q]−=F+−(p)δp,q.Following27,we modify the Gauss law generators asˆG ±(p)−→ˆ˜G±(p)=¯h p˜d2πρN(±p)that coincides with19.The modified Gauss law generators have vanishing vacuum expectation values,vac,A(t)|ˆ˜G±(p,t)|vac,A(t) =0.This justifies the definition20.For the zero componentˆG0,the vacuum expectation valuevac,A(t)|ˆG0|vac,A(t) =−12(e+η++e−η−)=1The quantum theory consistently describing the dynamics of the CSM should be definitely compatible with20.The corresponding quantum Hamiltonian is then defined by the conditions[ˆ˜H,ˆ˜G±(p)]−=0(p>0)(29)which specify thatˆ˜H must be invariant under the modified topologically trivial gauge transformations generated byˆ˜G±(p).We have in29a system of non-homogeneous equations in the Lagrange multipliersˆv H,±which become operators at the quantum level.The solution of these equations isˆv H,±(p)=¯hp2{pd4π¯h)2α∓p}.Substituting this expression forˆv H,±(p)into the quantum counterpart of28,on the physical states |phys;A we obtain1L2¯h2 p>0(d dα−p−1dαp,˜ddα±by˜d2L ˆπ2b−1dαp,˜d2Lˆπ2b+V(ρN;ρN),whereV(ρN;ρN)≡e2+Lp2ρN(−p)ρN(p)is the energy of the Coulomb current-current interaction.In order to make the dependence on N for the Hamiltonian more obvious,let us representρN asρN=12(1−N)σ,whereρ≡ρ1=ρ++ρ−,σ≡ρ−1=ρ+−ρ−,and[ρ(p),ρ(q)]−=[σ(p),σ(q)]−=0,[σ(p),ρ(q)]−=2pδp,−q.Then the Coulomb interaction energy takes the formV(ρN;ρN)=14(1−N)2V(σ;σ)+12Lˆπ2b+V(ρ;ρ).For N=−1,the momentum space electric charge density operator isσ(p)andˆ˜H EM =12π¯h:[e+b L2π¯h]+n,ˆψ+→exp{i2πn2π¯h ]→[e−b LLx}ˆψ−.The action of the topologically nontrivial gauge transformations on the states can be represented by the operatorsU n=exp{−i2π¯h ]−2πd[e+b L nρN(n)and U0is given by17.To identify the gauge transformation as belonging to the n th topological class we use the index n in31.The case n=0corresponds to the topologically trivial gauge transformations.The topologically nontrivial gauge transformation operators satisfy the same composition law as the topologically trivial ones.The modified operators are˜U n =exp{−i¯hˆTb})n|phys;A .Among all states|phys;A one may identify the eigenstates of the operators of the physical variables.The action of the topologically nontrivial gauge transformations on such states may, generally speaking,change only the phase of these states by a C–number,since with any gauge transformations both topologically trivial and nontrivial,the operators of the physical variables and the observables cannot be ing|phys;θ to designate these physical states,we haveexp{∓i¯h ˆTb})n|phys;A(so calledθ–states[26,27]),where|phys;A is an arbitrary physical state from20.In one dimension the parameterθis related to a constant background electricfield.To show this, let us introduce states which are invariant even against the topologically nontrivial gauge transfor-mations.Recalling that[e+b L2π¯h]θ}|phys;θ .(32)The new states|phys continue to be annihilated byˆ˜G±(p),and are also invariant under the topo-logically nontrivial gauge transformations.The electromagnetic part of the Hamiltonian transforms asˆHEM→exp{i[e+b L2π¯h]θ}=12L¯h2 p>0[˜d dα−p]+,i.e.in the new Hamiltonian the momentumˆπb is supplemented by the electricfield strength Eθ≡e+The condition34can be then rewritten as a system of linear equations in(β0,β±).We can easilyfind a solution of these equations,which gives us(β0,β±)as functions of[e+b L2π¯h}.However,these constants are irrelevant for our consideration and we neglect them.Finding(β0,β±)from34and substituting them into the expression33,on the physical states we obtainˆ˜H|phys;A =ˆHphys|phys;AwhereˆH phys =ˆH physF+ˆH physEM,ˆH physF=ˆH0,++ˆH0,−−1L¯h(1+N2)([e+b LL ¯h[e+b L2L ˆπ2b+V(ρN;ρN)+e2+L2π¯h] p∈Z p=0(−1)p 24(1−N2)2([e+b LL¯h p>0|λεp,±|−sρs±(−p)ρs±(p).Eqs.35and36give us a physical Hamiltonian invariant under both topologically trivial andnontrivial gauge transformations,ˆH physF andˆH physEMbeing invariant separately.The last two terms in35make invariant the free fermionic part of the Hamiltonian,while the ones in36the electromagnetic part.For N=±1,the last two terms in36vanish.These terms are therefore caused by the anomaly and represent new types of interaction which are absent in the nonanomalous models.The new interactions admit the following interpretation.Let us combine the last term in36with the kinetic part of the electromagnetic Hamiltonian,then124(1−N2)2([e+b L2L2 L/2−L/2dx(ˆπb−L E(x))2,i.e.the momentumˆπb is supplemented by the linearly rising electricfield strengthE(x)≡−e+2π¯h].As in four-dimensional models of a relativistic particle moving in an externalfield,we may define a generalized momentum operator in the formˆ˜πb(x)≡ˆπb−L E(x).The commutation relations for ˆ˜πb are[ˆ˜πb (x ),ˆ˜πb (y )]−=i (1−N 2)e 2+LL(1−N 2)[e +b L2L 2ˆ˜π2b→14π2(1−N 2)[e +b Lp 2ρN (p )=−e 2+L2p 2ρbgrd ·ρN (p ).It is just the background linearly rising electric field that couples b to the fermionic physical degreesof freedom in the Coulomb interaction.As a consequence,the eigenstates of the physical Hamiltonian are not a direct product of the purely fermionic Fock states and wave functionals of b .This is a common feature of gauge theories with anomaly.That the Hilbert space in such theories is not a tensor product of the Hilbert space for a gauge field and the fixed Hilbert space for fermions was shown in [6],[7].The background charge interpretation is related to the definition of the Fock vacuum.The definition given in Eqs.10-11depends on [e +b L2π¯h]is fixed.The values of the gauge field in regions of different [e +b L2π¯h]changes,then there is a nontrivial spectral flow,i.e.some of energy levels of the first quantized fermionic Hamiltonians cross zero and change sign.This means that the definition of the Fock vacuum changes.The charge operators ˆQ ±also change.Let :ˆQ (0)±:be charge operators defined in the region where[e +b L 2π¯h]the charge operators become :ˆQ (0)±:∓e ±[e ±b L。

小学上册英语第四单元真题试卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.What do we call a young ostrich?A. ChickB. CalfC. KitD. Fawn答案:A.Chick2.We are going to a ________ (音乐会).3.The starfish can regenerate lost ______ (部分).4.The boy plays the ________.5.My _______ (兔子) is curious about everything.6.The __________ is a famous area known for its art.7.My uncle shares his __________ (知识) about fishing.8.The __________ (历史的轮回) reminds us of the cyclical nature of events.9.What is the term for the movement of the Earth around the sun?A. RotationB. RevolutionC. OrbitD. Spin答案: B10.Which planet is known as the Red Planet?A. EarthB. MarsC. JupiterD. Saturn答案:B11.The ______ is the path of the Earth around the sun.12.Asteroids are mostly found in the _______ belt.13.My favorite fruit is ________ (葡萄) in summer.14.The ancient city of Pompeii was buried by the eruption of ______ (维苏威火山).15.What shape has three sides?A. SquareB. TriangleC. CircleD. Rectangle答案: B16.My pet fish swims around its ______ (鱼缸).17.What do you call the process of learning and gaining knowledge?A. EducationB. RecreationC. VacationD. Celebration答案:A18.The country known for its ancient ruins and temples is ________ (以古代遗址和庙宇闻名的国家是________).19.根据图片把下列单词补充完整。

高中英语世界著名科学家单选题50题1. Albert Einstein was born in ____.A. the United StatesB. GermanyC. FranceD. England答案：B。

解析：Albert Einstein（阿尔伯特·爱因斯坦）出生于德国。

本题主要考查对著名科学家爱因斯坦国籍相关的词汇知识。

在这几个选项中，the United States是美国，France是法国，England是英国，而爱因斯坦出生于德国，所以选B。

2. Isaac Newton is famous for his discovery of ____.A. electricityB. gravityC. radioactivityD. relativity答案：B。

解析：Isaac Newton 艾萨克·牛顿）以发现万有引力gravity）而闻名。

electricity是电，radioactivity是放射性，relativity 是相对论，这些都不是牛顿的主要发现，所以根据对牛顿主要成就的了解，选择B。

3. Marie Curie was the first woman to win ____ Nobel Prizes.A. oneB. twoC. threeD. four答案：B。

解析：Marie Curie 居里夫人）是第一位获得两项诺贝尔奖的女性。

这题主要考查数字相关的词汇以及对居里夫人成就的了解，她在放射性研究等方面的贡献使她两次获得诺贝尔奖，所以选B。

4. Thomas Edison is well - known for his invention of ____.A. the telephoneB. the light bulbC. the steam engineD. the computer答案：B。

解析：Thomas Edison（ 托马斯·爱迪生）以发明电灯（the light bulb）而闻名。

小学上册英语第5单元测验卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1. (46) is located at the equator. The ____2.The boiling point of a substance is the temperature at which it changes from ______ to gas.3.Chemical reactions can be classified as synthesis, decomposition, or ______.4.They are going to the _____ (公园).5.We draw pictures with ___ (crayons).6.Which is the smallest unit of life?A. OrganB. TissueC. CellD. Organism7.The Earth's mantle is made up of ______ rock.8.Which of these is a type of flower?A. OakB. RoseC. PineD. Maple9.I found a _____ (penny/dime) on the ground.10.My ________ (姑姑) is a great cook and loves to make dinner.11.What is 20 ÷ 4?A. 4B. 5C. 6D. 712.My friend is very __________ (积极向上).13.I watched a _______ (小蛇) slither by.14.The teacher gives us ______ (homework) every day.15.I can ___ (swim) in the pool.16.Which of these is a sweet treat?A. CakeB. BreadC. RiceD. Soup17.The elephant is the largest land ______ (动物).18.She is _____ (writing) in her notebook.19.The _____ (自然) cycle of plants is fascinating to study.20.The _______ (The Treaty of Tordesillas) divided the New World between Spain and Portugal.21.What do you call a group of lions?A. PackB. SchoolC. PrideD. FlockC22.What do we call the time it takes for the Earth to go around the sun?A. DayB. MonthC. YearD. WeekC23.The _______ can be very beneficial for your health.24.The boy plays the ________.25.Each plant has a specific _____ (需求).26.The pizza is very _______ (delicious).27.The __________ of a substance can change with temperature.28.The Andromeda Galaxy is moving towards the ______.29.My dog loves to dig _______ (洞) in the sand.30.My dad works as a _______ (工程师).31.The _____ (小鹿) is very graceful.32.The Great Wall of China was built to protect against _______.33.The seal barks loudly on the _________. (岩石)34.What do you call the person who writes books?A. AuthorB. EditorC. PublisherD. ReaderA35.Which animal is famous for its long migrations?A. ElephantB. SalmonC. LionD. TigerB36. A star will die after exhausting its ______.37.The rabbit has soft ______.38.My brother loves to go to ____ (concerts).39.The __________ is a famous mountain range in Europe.40. A _______ can measure the temperature of a gas.41.What do you call a young penguin?A. ChickB. PupC. CalfD. Kit42.I enjoy making ________ (生日蛋糕) for friends.43.The chemical formula for cetyl alcohol is ______.44.When I grow up, I want to have a ________ (赛车). It goes very ________ (快).45.The capital of Honduras is __________.46.What do we use to write on paper?A. BrushB. PencilC. EraserD. RulerB47.The chemical symbol for europium is _____.48.My sister is a ______. She loves to create art installations.49.What do you call a story that teaches a lesson?A. FableB. MythC. FairytaleD. NovelA50.The water is ________ (清澈).51.What is the main ingredient in aBLT sandwich?A. ChickenB. BaconC. TurkeyD. Ham52.The process of separating mixtures based on particle size is called _____.53.The ________ (生态影响监测) keeps track of changes.54.What is the opposite of sad?A. HappyB. AngryC. TiredD. BoredA55.What type of animal is a dolphin?A. FishB. ReptileC. MammalD. Amphibian56.What is the term for a baby kangaroo?A. CubB. KidC. JoeyD. CalfC57.My grandmother is my favorite _______ because she tells stories.58.The ________ (sweater) is warm and cozy.59.What is the primary color made by mixing red and blue?A. GreenB. PurpleC. OrangeD. Brown答案:B60.Which one is a vegetable?A. AppleB. CarrotC. BananaD. Grape61.The main use of hydrochloric acid is in _____.62.I think having a pet teaches us about __________.63.What is the process of changing from a gas to a solid called?A. DepositionB. SublimationC. CondensationD. EvaporationA64. A _____ is an area of land that rises sharply.65.The ocean is ________ (宁静).66.What is the opposite of empty?A. FullB. HeavyC. LightD. DeepA67.What is the common name for the substance that falls from the sky during rain?A. SnowB. HailC. WaterD. IceC68.We visit the ______ (自然博物馆) to learn about ecosystems.69.The chemical symbol for hafnium is __________.70.The hawk is known for its keen ______ (视力).71.My cousin is very . (我的表兄弟/表姐妹很。

小学下册英语原题[含答案]英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.What is 15 9?A. 4B. 5C. 6D. 7答案:C2.I have one ________ and two cats.3.I have a collection of ______ (玩具车) that I love to race with my friends.4.The __________ (历史的交流) enhances understanding.5.The _____ (teacher/student) is nice.6.Where do birds build their nests?A. In the waterB. In the groundC. In treesD. In caves7.What is the largest ocean on Earth?A. Atlantic OceanB. Indian OceanC. Arctic OceanD. Pacific Ocean答案:D8.The _____ is known for its beautiful rings.9.What do you call a person who studies insects?A. EntomologistB. BiologistC. ZoologistD. Botanist答案:A10.War was a period of tension between the ________ (美国和苏联). The Cold11.What is 3 x 3?A. 6B. 7C. 8D. 912.She has a beautiful ________.13.My favorite fruit is ________ (苹果).14.I enjoy _______ when it rains.15.The chemical formula for ethyl alcohol is __________.16.I have a ___ (story/book) to tell you.17.What do we call a person who writes poems?A. NovelistB. PoetC. AuthorD. Lyricist18.The teacher, ______ (老师), gives us projects to work on.19.What do we call a young porcupine?A. SpikeletB. PupC. KitD. Calf答案:C Kit20. A __________ (植物园) showcases many types of plants.21.What is the term for a written account of someone's life written by someone else?A. BiographyB. MemoirC. AutobiographyD. Novel答案:A22.What color is the sky on a clear day?A. GreenB. BlueC. RedD. Yellow23.How many colors are in a rainbow?A. 6B. 7C. 8D. 5答案:B24.My brother is a ______. He enjoys playing basketball.25.I want to be a ________.26.My hamster has a _______ (舒适的) cage.27.The sloth moves very _________. (慢)28.This toy ____ can jump really high! (玩具名称)29.Cleopatra was the last pharaoh of ________.30. A butterfly starts as a ______ (幼虫).31.What do we call the main ingredient in bread?A. FlourB. SugarC. WaterD. Yeast答案:A Flour32.The chemical formula for iron chloride is _______.33.I can ________ my toys.34. A reaction that absorbs heat is ______.35.How many hearts does an octopus have?A. OneB. TwoC. ThreeD. Four36.The pizza is _____ (hot/cold).37.What is the name of the desert in northern Africa?A. SaharaB. GobiC. KalahariD. Mojave38.The _____ (树木) in the park offer a place to play and picnic.39. A lever can help lift a ______.40.What is the term for a person who hunts for treasure?A. AdventurerB. ProspectorC. ArchaeologistD. Explorer答案:B41.I have a ______ (相机) to take pictures of my friends and family. It helps me capture ______ (回忆).42.What do we call a person who studies politics?A. PoliticianB. Political ScientistC. ActivistD. All of the above43.She is ________ a book.44.Star light takes millions of years to reach ______.45.What is the sound a duck makes?A. QuackB. MooC. BaaD. Roar答案:A46.The ______ (小鹿) grazes in the meadow, enjoying the warm sun.47.I saw a ________ in my friend's yard.48.My brother is my adventurous _______ who tries new things.49.My friend enjoys participating in ____ (debates).50.My cousin loves to __________. (唱歌)51.What do we call a person who studies the weather?A. MeteorologistB. ClimatologistC. AstronomerD. Geologist答案:A52.I put my _______ (shoes) by the door.53.What is the main color of the sun?A. BlueB. YellowC. RedD. Green54.What do we call the person who writes books?A. AuthorB. ArtistC. MusicianD. Actor55.We enjoy _____ (reading) stories.56.The __________ marks the boundary between the northern and southern hemispheres.57.The solid phase of water is known as __________.58.What do we call the process of water turning into ice?A. FreezingB. MeltingC. EvaporatingD. Boiling答案:A Freezing59.What color do you get when you mix red and white?A. BlueB. PinkC. PurpleD. Green60.We go _____ (camping) in the forest.61.What do you call a person who teaches students?A. DoctorB. TeacherC. ScientistD. Engineer62.My brother is a ______. He enjoys building robots.63.The baby is _____ (crying/laughing).64. A chemical reaction can change the physical ______.65.What is the name of the famous British author known for her novels about wizards?A. J.K. RowlingB. J.R.R. TolkienC. C.S. LewisD. Philip Pullman答案:A66.Gravity is the force that pulls objects ______.67.The __________ is a large, grassy plain found in North America.68.I have a toy _______ that can spin and twirl around.69.The chocolate is very ___ (sweet/bitter).70.How many legs does a dog have?A. TwoB. FourC. SixD. Eight71.What is the capital of Australia?A. SydneyB. MelbourneC. CanberraD. Brisbane答案:C72.My mom is my caring _______ who loves me dearly.73.What is the chemical symbol for sodium?A. NaB. SC. SnD. Si74.I have a great friendship with ____.75.What do you put on a sandwich?A. JellyB. Peanut butterC. CheeseD. All of the above76.I see a _____ frog by the pond. (small)77.The _____ (giraffe) is very tall.78.What do we call the science of classifying living organisms?A. ZoologyB. TaxonomyC. BotanyD. Ecology答案:B79.How many zeros are in one thousand?A. 1B. 2C. 3D. 480.He likes to _______ (draw/paint) pictures.81.I can design a _________ (玩具车) that can climb walls.82.The _______ (The Great Society) aimed to eliminate poverty and racial injustice.83.The country known for its multicultural societies is ________ (以多元文化社会闻名的国家是________).84. A ______ is a rule that describes a pattern in nature.85.The _____ (夜空) is filled with stars.86.I have a special ______ (玩具车) that I keep on my desk.87.The __________ is the part of the plant that develops into fruit.88.My favorite game to play during recess is ________ (跳绳) with my friends.89.What do we call a place where you can see many different types of animals?A. FarmB. ZooC. ParkD. Aquarium90.My aunt is a ______. She loves to paint.91.The sun is shining ___. (brightly)92.What do you call a baby cat?A. PuppyB. KittenC. CubD. Foal93.What do you use to cut paper?A. KnifeB. ScissorsC. GlueD. Tape答案:B94.What do you call the tall grass that grows in water?A. TreeB. ShrubC. ReedD. Moss答案:C95.The Great Wall of China was built over many ________.96.What is the name of the first President of the USA?A. Abraham LincolnB. George WashingtonC. Thomas JeffersonD. John Adams答案:B97.The __________ (历史的丰厚底蕴) inform practices.98. A bee's role as a pollinator is essential for many plants and ________________ (作物).99.In a chemical equation, the substances on the left side are called ______.100.Which planet is known as the Red Planet?A. EarthB. MarsC. JupiterD. Saturn。

第五讲：Lagrangian and Hamiltonian这里我想揭开 Hamiltonian 的神秘面纱， 还其本来面目： Hamiltonian 可从 Lagrangian 推出； 但有了 Hamiltonian 后，确实省事多了。

下面我只是从形式上导出两者的关系，这不能算作证明。

我比较注重的是帮大家增进理解。

希望获得严格证明的可去听六堂的课或研究 Kamien and Schwartz 的书。

还是用上一讲的 one control variable and one state variable 的模型。

0(4.1) Max ()(4.2) Subject to: (;)(4.3) (;)0, with (0) given.t U x e dt zg x z f x z z ρ∞−=≥∫(4.2)和(4.3)对每一时刻 t 都要成立。

为了便于写 Lagrangian, 我们可将 (4.2) 改写为: ()()(4.2a) Subject to: (();())0z t dt z t g x t z t dt+−−= 这里便是不严谨之处。

我们时而将 dt 看作是趋向于零的(比如在上式中), 时而把它当作某个固定的小数(比如在下面的推导中); 看我们需要而定 (这有点象牛顿-莱布尼兹开创微积分时的做法，微积分只有在εδ−语言之后才能严格起来 )。

现在就这个优化问题，写下 Lagrangian ， 把积分号想象成求和号。

[]000000()()(())()(();()()(();())()()(())()(();()()(();()()() ()()t t t t t t z t dt z t L U x t e dt t g x t z t e dt t f x t z t e dt dt z t dt z t U x t t g x t z t t f x t z t e dt t e dt dtz t dt z H t e dt t ρρρρρρμλμλμμ∞∞∞−−−∞∞−−∞−+−⎡⎤=+−+⎢⎥⎣⎦+−=++−+−=−∫∫∫∫∫∫0()t t e dt dtρ∞−∫[说明] 首先，()t μ是相对于（4.2a ）时间 t 的限制条件的拉格朗日乘子 （注意我们给()t μ乘上了t e dt ρ−，这和第三讲中将拉格朗日乘子乘上t β是一个道理）。

小学上册英语第2单元综合卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.Martin Luther King Jr. is known for his role in the ________ rights movement.2., I take my ________ (玩具名) to the park. I like to ________ (动词) with my friends there. We have a lot of ________ (名词) together. Sometime3.The capital of Russia is ________ (俄罗斯的首都是________).4. A reaction that produces a gas is indicative of a ______ change.5.Which country is famous for its pyramids?A. GreeceB. EgyptC. ItalyD. ChinaB6.This ________ (玩具) is a great way to relax.7.My favorite season is ________ (春天) because of the flowers.8.I love watching animals at the zoo. My favorite animal is __________.9. A planetarium is a building with a dome-shaped ceiling that shows ______ of the night sky.10.The _______ of the Earth protects us from harmful solar radiation.11.I found a _______ (小松鼠) raiding the bird feeder.12.The Earth's surface is shaped by both internal and ______ processes.13. A _____ (猫) loves to chase after string.14.The capital of Togo is ________ (洛美).15.The _______ (The Persian Gulf War) liberated Kuwait from Iraqi occupation.16.The chemical symbol for carbon is ______.17.What is the name of the famous giant redwood tree in California?A. General ShermanB. Redwood National ParkC. SequoiaD. Giant Sequoia18.The fish are swimming in the ________.19.The _____ (basket) is full of fruits.20.The substance that is dissolved in a solution is called a ______.21.The ______ of a tree is often wider than its trunk. (树的冠层通常比树干更宽。