Lecture Notes in Classical Mechanics

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LectureNotesinClassicalMechanics(80751) RazKupferman InstituteofMathematics TheHebrewUniversity July14,2008Contents 1 Preliminaries 1 1.1 Vectorcalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Dimensionalanalysis . . . . . . . . . . . . . . . . . . . . . . . . 5 2 NewtonianMechanics 15 2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Dynamics: Newton’slaws . . . . . . . . . . . . . . . . . . . . . 18 2.3 Workandmechanicalenergy . . . . . . . . . . . . . . . . . . . . 23 2.4 Angularmomentumandtorque . . . . . . . . . . . . . . . . . . . 26 2.5 Systemsofpointparticles . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Whatismissing? . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 d’Alembert’sprincipleofvirtualwork . . . . . . . . . . . . . . . 37 2.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 VariationalCalculus 45 3.1 Thebrachistochrone . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Functionalsovernormedspaces . . . . . . . . . . . . . . . . . . 47 3.3 Functionalderivatives . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Necessaryconditionsforalocalextremum . . . . . . . . . . . . . 50 3.5 TheEuler-Lagrangeequations . . . . . . . . . . . . . . . . . . . 52ii CONTENTS 4 Lagrangianmechanics 55 4.1 Hamilton’sprinciple . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Conservationtheorems . . . . . . . . . . . . . . . . . . . . . . . 56 5 Thetwo-bodycentralforceproblem 59 5.1 Reductiontoaone-particlesystem . . . . . . . . . . . . . . . . . 59 5.2 Analysisofthereducedone-particleproblem . . . . . . . . . . . 61 5.3 Classificationoforbits . . . . . . . . . . . . . . . . . . . . . . . 64 5.4 Orbitequations . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.5 Gravitationalsystems . . . . . . . . . . . . . . . . . . . . . . . . 67 6 Smalloscillations 71 6.1 Equilibriaanddeviations . . . . . . . . . . . . . . . . . . . . . . 71 6.2 Normalmodeequations . . . . . . . . . . . . . . . . . . . . . . . 73 6.3 Oscillationsofatri-atomicmolecule . . . . . . . . . . . . . . . . 74 7 Hamiltonianmechanics 77 7.1 TheLegendretransformation . . . . . . . . . . . . . . . . . . . . 77 7.2 Hamilton’sequations . . . . . . . . . . . . . . . . . . . . . . . . 79 7.3 Symmetriesandconservationlaws . . . . . . . . . . . . . . . . . 86 7.4 Derivationthroughavariationalprinciple . . . . . . . . . . . . . 87 7.5 Finalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8 Canonicaltransformations 89 8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2 Canonicaltransformations . . . . . . . . . . . . . . . . . . . . . 90 8.3 Choicesofgeneratingfunctions . . . . . . . . . . . . . . . . . . 91 8.4 Thesymplecticapproach . . . . . . . . . . . . . . . . . . . . . . 94 8.5 Poissonbrackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.6 Poissonbracketformulationofmechanics . . . . . . . . . . . . . 99 8.7 Differentiabletransformations . . . . . . . . . . . . . . . . . . . 99Chapter1 Preliminaries 1.1 Vectorcalculus 3 According to classical physics, “reality” takes place in a product space R ×R, 3 whereR represents space andR represents time. The notions of space and time are axiomatic in classical physics, meaning that they do not deserve a definition. (In relativistic physics, the notions of space and time are intermingled, and one ratherspeaksaboutafourdimensionalspace-time.) Because the physical space is a three-dimensional vector space, we will have to 3 dealextensivelywithvectorsinR . Wewilldenotevectorsbyboldfacecharacters, e.g., a,b,c. After choosing an an orthonormal basis, the entries, or components 3 ofavectorinR arecommonlydenotedby a = (a ,a ,a ). 1 2 3 Attimeswewillusedifferentnotations,suchasr = (x,y,z). 3 Leta andb be two vectors inR . Their scalar product, or dot product is a real numberdefinedby 3 a ·b = a b +a b +a b = ab. 1 1 2 2 3 3 i i i=1 Physicists often adopt a notational convention, theEinsteinsummationconven- tion, whereby indexes that appear twice in an expression are summed over, with- outneedofthesummationsign. Thatis, a ·b = ab. i i2 Chapter1 While confusing at first, this notation ends up being very useful. Of course, what 3 physicistscallascalarproductisnothingbutthestandardinnerproductinR . Thelength,ormagnitudeofavectora,isitsEuclideannorm, √ 1/2 a = (a ·a) = aa. i i The scalar product of two vectors, can be attributed a geometric meaning that involvestheangle,θ,betweenthetwovectors, a ·b = ab cosθ. Two (non-zero) vectors are said to be orthogonal (denoted a ⊥ b) if their dot productvanishes,i.e.,ifθ = ±π/2. 3 WewilloftendenotethestandardbasisinR by e = (1,0,0), e = (0,1,0), and e = (0,0,1). 1 2 3 Thus,avectorcanbealsowrittenas a = a e +a e +a e = ae. 1 1 2 2 3 3 i i Othertimes,wewillratherdenotee =xˆ,e =yˆ,ande =zˆ. 1 2 3 Proposition 1.1 Thedotproductsatisfiesthefollowingproperties: Itiscommutative,a ·b =b ·a. " Itisbilinear,(αa) ·b = α(a ·b). Itisdistributive,a ·(b+c) =a ·b+a ·c. TheCauchy-Schwarzinequality, a ·b≤ab. We define another product between pairs of vectors, the cross product, or the vector product. Unlike the dot product, the cross product results in a vector. It canbedefinedinseveralways,forexample, " " " " e e e " " 1 2 3 " " " " a×b = a a a = (a b −a b )e +(a b −a b )e +(a b −a b )e . " 1 2 3" 2 3 3 2 1 3 1 1 3 2 1 2 2 1 3 " " " " b b b 1 2 3 Theotherwayistodefinethecrossproductsofallpairsofbasisvectors,e.g., e ×e = 0, e ×e =e , etc., 1 1 1 2 3Preliminaries 3 and impose bilinearity and distributivity. A third definition introduces the Levi- Civitatensor,    +1 if(i, j,k)is(1,2,3),(2,3,1),or(3,1,2)     = −1 if(i, j,k)is(1,3,2),(2,1,3),or(3,2,1) .  ijk      0 otherwise. Then,thecrossproducttakesthesimpleform, a×b = ea b . ijk i j k A very useful property of the Levi-Civita tensor (easily checked by explicit sub- stitution)is = δ δ −δ δ . (1.1) ijk imn jm kn jn km The cross product also has a geometric interpretation. First, we claim that for everya,b, a ·(a×b) =b ·(a×b) = 0, thatis,thecrossproductoftwovectorsisperpendiculartoboth. Whilethiscanbe verifiedbya(relatively)tedioussubstitution,thecleanwayofderivingthisresult isusingourindexnotation,e.g., a ·(a×b) = (a)( a b) = aa b. i ikl k l ikl i k l Now, changessignwhenapairofindexesisswitched,hence, ikl a ·(a×b) =− aa b =− aa b, kil i k l ikl i k l where we have just renamed the summation indexes i→ k and k → i. Thus, this tripleproductequalsminusitself,henceitiszero. Whataboutitsmagnitude? Notethat 2 a×b = (a×b) ·(a×b) = a b a b . ijk j k imn m n Usingtheproductformula(1.1)fortheLevi-Civitatensor, 2 a×b = (δ δ −δ δ )a b a b jm kn jn km j k m n 2 2 2 2 2 2 = a b a b −a b a b = a b −(a ·b) = a b (1−cos θ), j k j k j k k j4 Chapter1 Figure1.1: Visualizationofthecrossproduct. i.e., a×b = ab sinθ. Thus, the cross product of two vectors is a vector perpendicular to both, whose magnitude equals to the area of the parallelepiped formed by the two vectors. Its directionisdeterminedbytherighthandrule(seeFigure1.1). Proposition 1.2 Thecrossproductsatisfiesthefollowingalgebraicproperties: Anti-symmetry: a×b =−b×a. " Distributivity: a×(b+c) =a×b+a×c. Bilinearity,(αa)×b = α(a×b). Itisnotassociative. % Jacobi’sidentity: a×(b×c)+c×(a×b)+b×(c×a) = 0. & The vector triple product, or “BAC minus CAB” formula: a× (b× c) = b(a ·c)−c(a ·b). ' Thescalartripleproductformula,a ·(b×c) =b ·(c×a) =c ·(a×b). ( Exercise 1.1 Proveit.Preliminaries 5 1.2 Dimensionalanalysis Units of measurement All physical quantities are expressed in terms of num- bers, obtained through measurements, during which comparison is made with a standard, or, a unit of measurement. For example, the length of a ruler is mea- sured by comparison to a standard, say the meter; the mass of a rock is measured by comparison with a unit mass, say, the gram; the duration of the day is mea- sured by comparison with a unit time, say the time it takes a standard hour-glass to empty; the (mean) velocity of a car is measured by measuring the distance it went, say in inches, the time it took, say in milliseconds, and dividing those two numbers. Fundamental and derived units The units of measurements are divided into twocategories: fundamentalunitsandderivedunits. Supposewewanttostudy a class of phenomena, for example, the motion of bodies. We may list all the quantities which will ever be measured, and for certain of them choose units of measurement, which we will call fundamental units; the choice is arbitrary. For example, we may choose fundamental units of mass, length, and time (but also force, length and time). Derived units of measurements are based upon the fun- damental units via some method of measurement (perhaps only conceptual). For example,themeasurementofvelocity,whose(derived)unitsusesthe(fundamen- tal)unitsoflengthandtime. Systemsofunits A set of fundamental units that is sufficient for measuring the propertiesofaclassofphenomenaiscalledasystemofunits,forexample, cgs=centimeter,gram,second inmechanics. Classesofsystemsofunits Twosystemsofunitswhichdifferonlyinthemag- nitudeoftheirstandards,butnotintheirphysicalnatures,aresaidtobelongtothe sameclass. Forexample,allsystemsofunitswhichareoftheform unitoflength = cm/L unitofmass = gram/M unitoftime = second/T,6 Chapter1 with L,M,T 0, belong to the same class (the LMT class). Another class, the LFTclassconsistsofsystemsofunitsoftheform unitoflength = cm/L unitofforce = kg-f/F unitoftime = second/T. (For the time being, ignore the fact we have no clue what “force” means. All you need to understand, is that it is a physical quantity that can be measured with the appropriateapparatus.) Note, however, that while both classes, the LMT and LFT classes, form a system of units for a class of physical phenomena which we call mechanics, they are not asufficientsetofunitsifwewant,inaddition,tomeasure,say,temperature,oran electriccharge. Dimensions Suppose we choose a class of system of units, for example, the LMT class in mechanics, and suppose we change our system of units within the same class, by decreasing the length unit by a factor L, the mass unit by a factor M, and the time unit by a factor T (e.g., we use centimeters rather than meters, ounces rather than grams, and weeks rather than seconds). How will this affect themagnitudeoftheoutcomeofmeasurements? LengthmeasurementswillbemagnifiedbyafactorofL;massmeasurementswill bemagnifiedbyafactorof M;timemeasurementswillbemagnifiedbyafactorof −1 T;velocitymeasurementswillbemagnifiedbyafactorofLT ,andsoon. Every physical quantity will be magnified by a factor which depends on L,M,T. That is,everyphysicalquantityhasanassociatedfunctionof L,M,T,whichwecallits dimension. The dimension of mass density (mass per unit volume), for example, −3 is ML . A quantity which remains invariant under a change of units is called dimensionless,anditsdimensionisbydefinitionone. Notethatthedimensionof aphysicalquantitydependsontheclassofthesystemofunits ( Exercise 1.2 ConsidertheMVTclasswherethefundamentalunitsarethatof mass, velocity and time. What kind of experiment measures length? What is the dimensionoflength? Whatisthedimensionofmassdensity? Physics is usually not thought in terms of an axiomatic theory. One rather speaks about fundamental principles. The most fundamental principle in all branches ofphysicsisprobablythefollowing:Preliminaries 7 Thelawsofphysicsareinvariantunderthechoiceofsystems ofunits. Whatarelawsofphysics? Theyarerelationsbetweenmeasurablequantities. For example, a physical law states that the force of attraction between two physical bodies (which can be measured) is proportional to the masses of each of them (whichcanbemeasured),andinverselyproportionaltothesquareoftheirsepara- tion(whichcanbemeasuredindependently). Thefirstobservationtobemadeisthatifaphysicallawisoftheform expression1 = expression2, then both expression must have the same dimension, otherwise the identity could notholdindependentlyonthesystemofunits. The dimension is always a power-law monomial So far, all dimension func- tions were always power-law monomials. Is it a coincidence? Suppose we work within the LMT class, and we are interested in some quantity a. By assumption (thattheLMTclassiscomplete),thedimensionofaonlydependson L,M,T: a = f(L,M,T). What does it mean? That if we change units by dividing the units of length, mass and time by L , M , T , the measured value a will increase by f(L ,M ,T ). 1 1 1 1 1 1 1 Similarly,ifwechangeunitsbydividingtheunitsoflength,massandtimeby L , 2 M ,T ,themeasuredvaluea willincreaseby f(L ,M ,T ). Thatis, 2 2 2 2 2 2 a f(L ,M ,T ) 2 2 2 2 = . a f(L ,M ,T ) 1 1 1 1 The underlying assumption is that all systems of units within a given class are equivalent. Thus,wemaythinkofthesystem1astheoriginalsystem,andsystem 2 as obtained by decreasing the fundamental units by L /L , M /M , and T /T . 2 1 2 1 2 1 Then, a = a f(L /L ,M /M ,T /T ), 2 1 2 1 2 1 2 1 fromwhichimmediatelyfollowsthefunctionalequation ' ( f(L ,M ,T ) L M T 2 2 2 2 2 2 = f , , . f(L ,M ,T ) L M T 1 1 1 1 1 18 Chapter1 Whatcanbelearnedfromsuchafunctionalrelation? Assuming that the dimension function is smooth, we differentiate both sides with respectto L andset L = L = L, M = M = M,andT = T = T, 2 2 1 2 1 2 1 1 ∂f 1∂f α (L,M,T) = (1,1,1)≡ , f(L,M,T)∂L L∂L L fromwhichweconcludethat α f(L,M,T) = L g(M,T). Substitutingintothefunctionalequationweget ' ( g(M ,T ) M T 2 2 2 2 = g , , g(M ,T ) M T 1 1 1 1 andbythesameproceduregetthat β g(M,T) = M h(T). γ Repeatingthisforathirdtimewegeth(T) = cT ,i.e., α β γ f(L,M,T) =cL M T . Since f(1,1,1) = 1,weconcludethatc = 1. Quantities with independent dimensions The physical quantities a ,...,a 1 k are said to have independent dimensions if none of these quantities have a di- mension which can be presented as a power monomial of the dimensions of the remainingquantities. Itisperhapseasiertogivealogarithmicformulation. Inthe LMTsystem,forexample,everysetofphysicalquantities(a), i = 1,...,k,hasa i dimensionfunctionsoftheform, loga = α logL+β logM +γ logT 1 1 1 1 . . . . . = . loga = α logL+β logM +γ logT k k k k Thatis,thelogarithmsofdimensionsformavectorspacewithbasisvectorslogL, logM, logT. Independence of dimension is linear independence over this space.Preliminaries 9 Obviously, in this case there can be at most three physical quantities of indepen- dentdimension. Suppose now physical quantities a ,...,a and b ,...,b , such that the first k 1 k 1 m haveindependentdimensions,andallthemadditionalquantitieshavedimensions that depend on the dimensions of the first k quantities. The fact that the a have i independent dimensions implies that if we work, say, with an L L ...L system, 1 2 r thennecessarily,r≥ k andif      loga γ ··· γ logL      1 11 1r 1                . . . . .       .   . . .  .    =   , . . . . .                loga γ ··· γ logL k k1 kr r then the matrix of coefficients has full rank (i.e., k). This has an important impli- cation: itispossibletochangeunitsofmeasurementsuchthat,asaresult,asingle a changesitsvalue,whilealltheotherremainfixed. j Now we come to the b . Since their dimensions depend on the dimensions of the j a ,thenthereexistconstantsα ,suchthat j ij logb = α loga + ···+α loga 1 11 1 1k k . . . . . = . logb = α loga + ···+α loga . m m1 1 mk k Considernowthefollowingnewquantities, b 1 Π = 1 α α 11 1k a ...a 1 k . . . . . = . b m Π = . m α α m1 mk a ...a 1 k It is easy to see that these quantities are dimensionless; their value does not change,nomatterhowwechangeourunitsofmeasurement. Physicallaws Aphysicallawalwaysconsistsofarelationshipbetweenphysical quantities, c = f(a ,...,a ,b ,...,b ). (1.2) 1 k 1 m10 Chapter1 Herecisthequantitybeingdeterminedanda ,...,a ,b ,...,b areallthequan- 1 k 1 m tities it depends upon (as above, the a have independent dimensions). It is im- j portanttostressthataphysicallawisanequationthatrelatesbetweenmeasurable quantities. There exists a function f, such that if one measures the quantities a , j b ,andc,thenumbersthusobtainedsatisfytheprescribedequation. j We first claim that the dimension of c must depend on the dimension of the a . j Why is it so? Otherwise, we could perform a change of units of measurement, such that the value of c changes, while the values of all the a (and consequently j alltheb )remainunchanged. Thiswouldcontradicttheassumptionthat f isonly j afunctionofthosephysicalquantities. Thus,thereexistnumbersβ ,...,β ,such 1 k that β β 1 k c = a ...a , 1 k andthephysicalquantity c Π = β β 1 k a ...a 1 k isdimensionless. TheΠtheorem Usingourdefinitionsofdimensionlessquantities,wecanrewrite (1.2)asfollows: / 0 1 α α α α 11 1k m1 mk Π = f a ,...,a ,a ...a Π ,...,a ...a Π , 1 k 1 m 1 k 1 k β β 1 k a ...a 1 k andthiscanbebroughtintoanalternativeform, Π =F(a ,...,a ,Π ,...,Π ). 1 k 1 m That is, we rewrote (1.2) as a relation between k physical quantities of indepen- dentdimensionand mdimensionlessquantities,ononeside,andadimensionless quantity,ontheotherside. Now remember we can perform a change of units such that only a changes its 1 value. Thus,F cannot depend on a . By repeating this argument k times we end 1 upwiththeconclusionthat Π =F(Π ,...,Π ). 1 m This is known as the “Π-theorem”. A physical relationship between some dimen- sionalparameterandseveraldimensionalgoverningparameterscanberewrittenPreliminaries 11 as a relation between some dimensionless parameter and several dimensionless productsofthegoverningparameters. What is the immediate gain? A law that seemed to depend on k + m parameters reduces into a law that only depends on m parameters. This might be a huge reduction. Example: In Euclidean geometry it is known that the area S of a right triangle depends only on the length of its hypotenuse, c, and the magnitude φ of, say, the smallerofitsacuteangles. Thus, S = f(c,φ). For this problem we can do well with for only fundamental units those of length (an L system). In this case, c has dimension L, φ is dimensionless and S has 2 2 dimension L . By theΠ-theorem, the dimensionless quantityΠ = S/c can only dependonthedimensionlessparameterΠ = φ, 1 2 Π =F(Π ) ⇒ S = c F(φ). 1 WithoutcaringwhatthefunctionF actuallyis,wehaveatonce, 2 2 2 2 S = a F(φ), S = b F(φ), S +S = c F(φ), 1 2 1 fromwhichfollowsPythagoras’theorem. S c 2 b S 1 a 12 Chapter1 Example: A mass m is thrown vertically with velocity v. What is the maximal heighthitwillattain? Westartwith h = f(m,v,g). 2 Since h = L,m = M,v = L/T and g = L/T , we conclude from the Π-theoremthat h Π = = const. 1 2 v /g ( Exercise 1.3 A pendulum is a massive particle suspended on a string (which we assume to have negligible mass). If one perturbs the pendulum (for example byplacingitattheequilibriumpoint,butassigningitaninitialvelocity),thenthe pendulum performs a periodic motion. Suppose we want to find a law to predict the period, τ, of the motion. We start by making a list of all quantities that τ may dependupon: (1)themassmofoftheparticle,(2)theinitialvelocityv,(3)earth’s accelerationg,and(4)thelengthofthestring,+. IntheLMTsystemwehave L 2 3 L τ = T m = M v = g = + = L. 2 T T Whatcanyousayaboutthefunction τ = f(m,v,g,+) basedondimensionalanalysis? " Suppose now an experimentalist tells you that the period does not depend ontheinitialvelocity. Howwillyouranswerchange? Example: In an atomic explosion, a large amount of energy E is released within asmallregion,andastrongsphericalshockwavedevelopsatthepointofdetona- tion. In the early stages, the pressure behind the shock wave is huge compared to the atmospheric pressure, which is completely negligible. If r denotes the radius ofthewaveandt istime,theradiusisonlyexpectedtodependon r = f(t,E,ρ), whereρisthedensityoftheairatequilibrium(noshockwavewithoutairdensity, andothercharacteristicsofair,suchassoundwavespeedareirrelevantforshock waves).Preliminaries 13 ThedimensionsofthegoverningparametersintheLMTclassare 2 3 2 −2 −3 t = T E = ML T ρ = ML . It is easy to see that those are all independent, i.e., k = 3 and m = 0. The dimensionofr is L,sothat 1/5 ρ r Π = 1/5 2/5 E t is dimensionless, and depends on nothing, i.e., must be a constant. This means that −1/5 1/5 2/5 r = Cρ E t , or 1 1 2 logr = logC− logρ+ logE + logt. 5 5 5 Thus, if we plot logr versus logt, the slope should be 2/5, and the graph should intersectzeroatthevalue 1 1 logC− logρ+ logE. 5 5 Sinceρisknown,knowingtheconstantC wouldrevealtheenergyoftheblast. In the 1940’s the Americans performed nuclear tests, which were photographed by J. Mack. Those photographs were not classified. G.I. Taylor analyzed those pictures. HeknewthattheconstantC wasaboutO(1)(onecanfinditfromasmall explosion), and based on the above analysis published the value of the energy 21 (about10 ergs)whichcausedahugeembarrassment. 14 Chapter1Chapter2 NewtonianMechanics InthischapterwewillreviewthebasicsofNewton’s“old”classicalmechanics— oldinthesensethatitislessgeneralandformalthanthemorerecentformulations ofLagrangeandHamilton—classical,todistinguishfromquantummechanics. 2.1 Kinematics Mechanics primarily deals with the motion of so-called point particles. A point particle is a model of a physical object whose state is fully specified by its loca- 3 tion, the latter being a point inR . It is important to distinguish between models andreality. Thepointparticleisanidealization,whichcanonlybejustifiedempir- ically. Attimes,thesolarsystemcanbedescribedascomprisedofpointparticles, even though the sun and the planets are complex objects. In other instances, even a round tennis ball cannot be modeled as a point particle (e.g., if rotation is im- portant). Physics make predictions about how physical entities evolve in time. Given a point particle, its evolution is completely determined by its position r = (x,y,z), as function of time t, or its trajectory. Mathematically, the trajectory r(t) is a 3 pathinR . Forthetimebeing,wewillassumethatthetrajectoryisatleasttwice differentiable with respect to time; the justification for this assumption will be giveninthenextsection. Note that the definition of r assumes the choice of an origin, with a specific ori- entationoftheprincipalaxes. Inordertobeabletomeasurerwealsoneedaunit16 Chapter2 of length. Centimeters and meters are the most common choices (for those who have never heard of inches and feet...). Time also requires an origin and a unit of measurement,secondsbeingthemoststandardchoice. Therate-of-changeofthepositionr(t),isthevector dr v(t) = (t), dt which is called thevelocityvector of the point particle. Note that this is a vector identity,namely,v = (v ,v ,v ),with 1 2 3 dx dy dz v = , v = , and v = . 1 2 3 dt dt dt The measurement of velocity is somewhat “conceptual”. We imagine two mea- surements of position at time t and time t+Δt, and an approximation to velocity intermsofthechangeinrdividedbyΔt. TheactualvelocityisthelimitwhereΔt is infinitely small. Despite the fact that the exact velocity can never be measured, thisconceptualmeasurementsufficestoassignvelocitydimensionsofL/T,where L is the dimension of length and T the dimension of time. If we measure length inunitsofmetersandtimeinunitsofseconds,thentheratioofΔr/Δt isassigned a unit of meters/seconds. This unit has to be understood in the following way: the velocity was determined by a ratio of displacement and time interval, in an experimentsinwhichlengthwasmeasuredinmetersandtimeinseconds. Note that if we know the velocity as function of time, v(t), then the position can beretrievedbyintegration,butonlyuptoaconstanttranslation, 4 t r(t) =r(t )+ v(s)ds. 0 t 0 If the velocity is constant, v(t) = v, then the displacement is a linear function of time, r(t) =r(t )+v(t−t ). 0 0 Thetimederivativeofthevelocity,whichisthesecondderivativeofthetrajectory, iscalledtheaccelerationofthepointparticle,andisdenotedby 2 dv d r a(t) = (t) = (t). 2 dt dtNewtonianMechanics 17 2 2 Acceleration has dimensions of L/T , and is usually measured in units of m/sec (again, we don’t really mean that a meter is divided by the square of a second). Giventheacceleration,a(t),thevelocityisgivenby 4 t v(t) =v(t )+ a(s)ds, 0 t 0 andthepositionisgivenby 4 4 t s r(t) =r(t )+v(t )(t−t )+ a(τ)dτds. 0 0 0 t t 0 0 Whentheaccelerationisconstanta(t) = a,then v(t) =v(t )+a(t−t ), 0 0 fromwhichweobtainthetrajectory, 1 2 r(t) =r(t )+v(t )(t−t )+ a(t−t ) , 0 0 0 0 2 which requirestheknowledgeoftheinitialpositionandvelocityofthepointpar- ticle. Example: Itisanempiricalfactthatbodiesexperiencingonlytheforcesofgravity haveaconstantacceleration, m a = (0,0,−9.81) 2 sec regardlessoftheirmass. Hereitisassumedthatthez-axispointsupward,whereas the x,yplaneistangenttothesurfaceofearth. Supposeweareinaflyingballoon, and we throw a massive body from the point (0,0,100)m with initial velocity (3,0,10)m/sec. Whenandwherewillthebodyhittheground? Bytheaboveformula,settingt = 0, 0 1 2 r(t) = (0,0,100)+(3,0,10)t− (0,0,9.81)t . 2 Componentwise,wehave 9.81 2 x(t) = 3t, y(t) = 0, and z(t) = 100+10t− t . 2

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