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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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