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classicalmechanics is thebasis forall
ofphysics it's the basis ofall of
physicsnot only because it describes
the motionof objectslike particles and
mechanicalsystems andso forth but
because the basic frameworkthe basic
structureof allthe physics isbased on
the principles of classicalmechanics
the conservation of energy the
conservation of momentumthe principles
by whichall systems evolve in nature is
thesame set ofrules essentially
exactlythe same setof rules in a more
abstractin the more general setting
then the rules which govern how a simple
particle moves for example under the
influence ofgravity but in order to
understandthat we have tounderstand
the principles in afairly general
contextlet's begin with the veryvery
simplest kinds of systems thatwe can
think ofsystems that are so simplethat
in fact issimple than any real systems
in naturelaws of nature let's imagine
laws of nature whichare of themost
primitiveand simple kind acting on the
most primitive and simple systems that
wecan imagine I want you to first of
all supposethat timewhich evolves
continuouslyunder the my watch I see
the second hand goes around andaround
andaround that goes around continuously
the time can evolveand be any real
numbert I want you to imagine it only
occurs inbeats a stroboscopic world in
which you only lookat the world at
intervals of timewhich we couldto be a
thousandth of a second or it could be a
millionth of a secondlet's just take it
to be one second intervalsand ask how
in the stroboscopic world systems change
with timelet's also imaginea very very
simple set of systems systems whichare
sosimple that they only have a handful
of configurationsconfigurations mean
everything we need
knowabout the systemto characterizeit
completely so the simplest systemI can
thinkof would bea system that has only
two configurationsheads or tails a coin
lay according down on the table I don't
have a coin so we'll take this coffee
top hereI could putit down headsor I
couldput it down tailsat that if you
can tellthe differencefrom where
you're sittingbut I can tell the
differencethis is this is heads that's
tailsall right so we havea system then
it's characterized bytwo states two
states of being and we want to add to
those two statesa law a law of
evolutionin going fromone instant of
time to the nextinstant oftime from
one beat ofthe stroboscopic light to
the nextone what kind of laws are
allowablewhat kind of lawsthe basic
principles of physicsallow and what
kind of laws don't they allow that's
going to be ourfirst kind of question
for tonight
so our firstconcept isthe spaceof
states in this case just headsin tails
it's justtwo pointsheads and tails so
two points inan abstract spaceheads
and tailsit's calledthe phase space
it's called the phase spaceof a system
the space of possible states ofa system
and what do Imean by a state of a
systemI mean everything thatyou need
to know in order topredict whathappens
next
everything you needto know in order to
be ableto say withcertainty what the
next state ofthe systemwill be that's
called the phase spaceof thesystem in
thiscase just headsin tailswhat kind
of lawscan you imagine whatkind of
lawsof nature can you imagine forthis
extremely simple world it is the
simplest well I suppose youcould
imagine aslightly simpler worldyou can
imagine a worldwith onlyone state
heads not much can happen in that world
there's onlyone law of physics heads
goes to head since heads goesto heads
goes toheads it's extremely boring
nothing ever happens because there's
only one state so how could anything
happen but with two stateshas entails
you can have a richer variety of
possible laws of physicsone law let's
take the various varietiesof lawsof
physics wecan have one law might just
say you stay the same if the stateof
thesystem is heads at one instant of
time thenext instant oftime it willbe
heads
if it startstails it will stay tails
that's a very boring lawand let's graph
that law by drawing anarrow
if it starts headsthen it stays heads
let's just represent thatby drawing an
arrow from head to headand from tailto
tail the meaning of this arrowyou start
at the tailend of the arrow and you
followit and yousee thatit comes
aroundto the same pointthat stands for
the rule that saysthat a headsstays a
headsand in this case atail stays a
tail so
a dull law of nature thisstays that way
for endless amounts of time and this
stays that way forendless amounts of
timeanother possible law of nature
would be slightly moreinteresting
situationif you look at it at one
instantof timethe Nexen since it's the
opposite
heads goesto tailsbecause the heads
goes to tails
these are deterministiclaws
deterministic laws mean that if you know
what is happeningat one instantyou
know forever after you know everything
about the systeminfinitely into the
futurecompletely deterministic
classical mechanics has that nature to
itthat iscompletely deterministicso
and in factit isin acertain sense
always reversible but we'recoming for
that allright so to draw the graph
representing heads goes to tails and
Tails goes to heads we draw an arrow
fromheads to tailsand from tails -
headsand we readthat assaying thatif
youstart at heads you go to tail if you
start atpals you go to heads
what is the evolution of asystem under
this lawof natureif you start with
headsit's heads tails headstails heads
tailsheads tailsforever after these
are two consistent laws of physics in a
world ofonly two stateswhether we can
think ofmore lawsyes we can butbut
for the momentthese are two interesting
ones now how can we generalizethis we
cangeneralize thisfirst of allto
systems with moreStates to States is
not very many we couldhave a die die as
in dice die has six statesone two three
four five andsix and we could represent
them as pointssix points now we have a
large variety ofdifferent lawsof
physics thatwe could have for example
we could simply have we couldlabel
theseone two three four five andsix
which one is oneand which one is six is
not very importantbut the
there are six of them we could have a
law of physicswhich says 1 goes to 2 2
goesto 3 3 goes to4 4 goes to 5 5 goes
to 6 and 6 goes back to 1this would be
a complicatedmotion when thought of in
termsof adye you start with a1 upand
then it goesto the 2 up 3up and so
forthbut it's a relatively simple
picture when drawn in this manner here
you can imaginemore complicated laws of
physicswith six stateswell whether
they're morecomplicated or not is in
the eye of thebeholder butthere first
ofall we could have a similar lawwhere
instead of one goingto two one could go
to threethree could go to to two could
go to five one two three somethinglike
that that'sreally not verydifferent
than thiseach onegoes toa neighboring
one well noton neighboringone each one
goes toa next one and they cycle around
and one characteristic of such systems
is they will just cyclearound forever
andever andever
okay now you canhave more complicatedI
don't know if they'remore complicated
different lawsfor example you could
have a law which goes this way if you
start at one you go to two if you go to
twothen you go tothree and thenyou go
backto oneif you happen to start at
three note this isthree thisis five
rightfive six six six six one two three
four five if you startat three you go
to four if you start at four you go to
fiveback to three in this case
well that's somewhatsimilar tothis you
have two disconnectedcycles to do got
more complicatedthings here's another
one infact I think itwould takea long
timeto draw all thepossibilities one
goes backto 1 2 goes to 3 3 goesto 4
all goes back to 2
five goes tosix andsix goes back to
five rightthey're all acceptable laws
of physicswherever you happen to beyou
know exactly whereto go next
so it'sdeterministic it's deterministic
into the futuremeaning to say wherever
you start you know where youwill be
arbitrarily into the future if you start
here andyou go a hundred thousand times
you'll justwind upwriting up some
whereverwherever you start hereyou
stay thereso these arecompletely
deterministic into the futurebut
they're also completelydeterministic
into the pastwhich meansif you know
where you areyou know where you were
beforeif you find yourself over here
thenyou know in theprevious cycleor
the previousinstant you were over here
and so forthso you can trace your
motion either into the future or into
the past with complete confidenceabout
where you'll be no matter how far you go
that's thecharacter in principlein
principle if not in practice and
practice things get jumbled upand you
can't see themclearly enough and you
and you miss detailbut if you can see
the infinitely small detail ina system
and get every singlebit of physics
absolutely rightfor any classical
system theyare in this sense exactly
deterministic both into thepast and
into the future now what kind of laws of
physics do we not allow the kind of laws
ofphysics thatwe don't allow I can
best illustrate I think by drawing some
some possibilities andexplaining to you
why theyare not allowed here's a law of
physicsthat is not allowedby the
principles of classicalphysics three
statesthree states is perfectlyalright
nothing wrong with that but let's draw
some arrows- 3 2 goes to 3and 3 goes
back to 2 that wouldbe perfectly
alright by itselfwhat about 1 well
I could have one goes toone butI don't
want oneto go to one I want oneto go
totwo that'scompletely deterministic
into the futureif I start at one I go
to twoI go to two I go to threeI go
three backtwo to two back to threeI
always know where togo in thenext step
Ijust follow the arrow whereverit
happens tobe but what doesit failit
failsto be deterministic into the past
supposingI know that I'm at- then
where didI come from
I could have comefrom one or I could
have come fromthree so I cannot work my
way backwardwith uniquenessI can work
my way forward
I cannot work my way backward that's a
this is a lawof physicswhich is
irreversibleit wouldnot allowme to
run the laws of physics backwardit
would lead to an ambiguity every time I
were at to another law which is not
allowed by the principles ofclassical
mechanics or principles ofclassical
physicswould be basicallythe same
thing butwith the arrows turned in the
opposite directionall arrows reversed
here I have aproblem not going intothe
past but I havea problem going into the
future
let's say yeah supposing nowI find
myself a- and I want to gointo the
futureI don't know whetherto go into
thefuture by following this arrowto
one or this arrow to three there are two
arrows leadingout of two over hereone
of themgoes tothree one ofthem goes
toone nothingtells me which I wrote to
follow so it'snot deterministic into
the future I might randomly decide to go
from two tothree or randomly decide to
go from twoto one these are ruleseach
oneof which isdeterministic in one
direction but not the other
these are the sorts of thingswhich are
forbiddenby the principles of classical
mechanics
why are they forbidden bythe
principal's yeah which one yeahthat
wouldn't come I wouldn't know whereto
go fromthree you get stuck at three
yeahno good well you can't have a
dangling one you gotta know where to go
next
now you could let's see we could try to
put something like that in yeahokay so
Iwould say we go from 1 to2 from 2 to
3 2 32 3 23 now the problem with being
going backward I think yeah yeah yeah
right so one way or the other youget
stuckover these rules all right how do
we spot what's allowable andwhat's not
allowablewell it's very simple
at every configuration2 or 3 we should
have one in and one out
we should have aunique onein a unique
arrowin and a uniquearrow out one
arrow in to tellus where we came from
and one arrow out to tell us where we're
going
that's the characterof classical
physics uniqueness into the future
uniqueness into the past when
represented in terms of this very simple
andanalog worldanalog digital world of
our finite number ofstates then the
rules of physicsas we knowthem would
say everyconfiguration has one in arrow
in one out arrow now with thisrule of
course lifeis very boringbecause if
you only have a finitenumber of states
all it canhappen to you is you cycle
aroundendlessly among those number of
States always in the same orderyou can
haveslightly more interesting
situations there's no reason why the
number ofStates has to befinite you
could have a situation wherethere's an
infinite numberof possible state
a state correspondingto every integer
positive and negativea particle on a
position on a line where the position
could be any integer value and then a
simple law ofphysics wouldbe you go
from oneto thenext wherever you are
you goto the nextone eachpoint has
onen in one out I can't draw them
endlessly it'll take foreverbut each
markone in and one outthis of course
would also be arather boring law of
natureyou just hop fromone to thenext
to the next and next andnext forever
and ever andever but at least you
wouldn't be cyclingaround endlessly
again we can have add some morestates
on to this we could add this onif we
start on the linewe simply move off and
keep going foreverand ever andever but
if we startover here we cycle around so
we couldhave mixtures ofboth kinds of
thingssome states we cyclearound in
other stateswe move off toinfinity now
notice that someof these laws of nature
the phase space breaks upinto different
pieces which are connected among
themselves but notconnected to each
otherfor example even with justtwo
states we had two possibilitiesone like
this and onelike this in this case
where itbreaks upinto more thanone
more than one piece we have something
calleda conservation lawa conservation
lawis simply a memory ofwhere we
started
a conservation lawmeans thatsomething
is keptintact for all time some piece
of knowledge is keptintact for all time
anddoesn't changein this case we could
label this we could labela
configurationover here with a plusone
and a configurationover here with a
minus one
and then we could call we could invent a
variable plusone overhere and minus
one over here itnever changesif it's
plusone that stays plus one if it's
minusone it stays minus onethat's a
conservation lawsomething whichdoesn't
change with timeon the otherhand if we
hop from plus 1 to minus 1to plus1 to
minus 1we don't have a conservationlaw
conservation laws are alwaysassociated
with these kindof closedfamilies of
different trajectories in the phase
spacewhich don't mix witheach other
which rememberarm something about the
systemwhich might otherwise get mixed
upif everything gotall mixedtogether
so there's all kinds of possibilities
that are inherent in these deterministic
laws but always the condition isone in
line and one out line for every point
that can alsobe called information
conservationits information
conservationin exactly the sense that
that you never lose memory of where you
startedeither into thepast orinto the
futureif you know where you areat any
instant youknow where you came fromand
youknow where you'll be information
about where you are is conservednever
changes intothe past in the future
whereas ifyou have oneof these bad
laws laws which areforbidden by the
rulesof classical mechanics then you do
lose informationfor example if youfind
yourself over here you don't know
whether you camefrom here orfrom well
nothat's not quite right find yourself
over here you don'tknow whether you
came from here or over here so you lose
information information conservation is
perhaps the mostfundamental law of
basic classical physics thatyou don't
loseinformation about now why thatwhy
is thatso it's not writteninto the
laws of physics why they are what they
aremaybe someday we'll understand
Hyper laws
physics ormetal laws of physics or
deeper laws of physics which will tell
uswhy the laws of physicsare what they
are at the momentit's moreor less an
experimentalfact that all the known
lawsof physics fit into this class of
information conserving lawseven those
which are quantum mechanical but for our
subjectthis quarter we're only
interested in classical physicsall
right so that's thebasic set up if we
were interestedonly in this
stroboscopic world of discrete time
intervalsthe real world ofcourse is
more continuous than that supposingokay
so let's make the law somethinglike
this if the last two states were heads
headsthen it stays heads if the last
two states were heads tails then it goes
to heads if the last two with heads
tailsheads then it goesto tailsand if
the lasttwo with tails tailsthen it
goesto tailsI think that's a possible
possible law then youwould say I can't
tell from thefact thatit's heads where
it goes nextand indeed I can't but that
would justbe another way of saying that
a specificationof a heads by itself is
notwhat youwould call a statewhere
youwould call a state would be the
specificationof the previous last two
entriesbecause you need two entries to
tell you what happensnext now that
raises the question that that's very
veryimportant in classical mechanics
how much and whatexactly do you need to
knowto say what happens next
if the phase space is thespace of
thingsspace of possibilitiesbut always
in such a way thatthey tell you exactly
what happens nextwhat isit that you do
haveto knownext
so that brings us to continuous physics
let's take the motionof a particle
let's take the motionof aparticle is
it enough toknow where a particleis to
say what happens next let's hypothesize
thatthe generalization the continuous
time is that we needto know the exact
location of a particle along a line
we've runout of ink I'm afraid not
there is all rightso let's imagine the
motion of aparticle alonga line then
you mightthink that the analogof a
state is just alocation ofa particle
where is itbut is itenough toknow
where a particleis inorder to say what
happens nextnow whatelse do youneed
to knowits velocityin orderto know
where aparticle willbe nextyou need
toknow not onlywhere it is but how
fast it's movingyou need to know its
velocityso thatmeans the state in the
same sense that I used itthat whichyou
need to know inorder to know what
happensnext
does not just consistof the location of
a particle but you can sayit two ways
you needto know notjust the location
of theparticle but you need to know
also the previouslocation or better yet
whatcauses for what is equivalent to
knowingthe previouslocation the
velocityof velocitythat tells youthat
the phase spacethe spaceof states the
space of configurations is
two-dimensional notone-dimensional it's
not just a line it's a linethat
represents the position of theparticle
and a second line whichrepresents its
velocityeither to the left orto the
rightpositive velocity means moving to
the rightnegative velocity means moving
to the leftsupposing you're over here
where do you movenext
you stay the same placewhy you're at
the originand you have no velocitywhat
if you're over here so wecould justwe
could draw a littleloop here to say
thatyou come back tothe same place
what if you're overhere you stay the
same placebecause you're movingwith 0
velocity vertical axis isvelocity so
youcome backto the same placewhat if
you're over here where are youone
second laterlet's chop time up into
one-second intervalswhere are you next
somewhere to theright huh what if
you're over here now what if you're over
here move twiceas far orroughly twice
as farto the right
what is it down hereyou move to the
leftso we could fill up thisspace here
withlittle tiny arrows to show where
you movenext but notice- no way you
move nextyou haveto know not only
where you are in the sense of what X is
but you also have toknow the vertical
component namelywhat the velocity isso
that means theanalog the analog of a
point in the phasespace is a point in
the spacenot only of positions but also
velocities yeswhy
where that takeplace it meansno it
meansexactly what the gentlemanasked
me beforewhat if you had a law of
nature which tells you in orderto know
where tomove next you have toknow your
previous two entriesall right now one
way ofsaying it isall right in that
case I needto know the previoustwo
entriesand it just doesn't fallinto
this class of thingsor youcould say
that the space of configurations doesn't
just consist of a heads ora tailsbut
it consists of a pairof entries a pair
yeah that's rightwe meant we're not we
modelthat that'sright we need two axes
that that's right we need an axisfor
the presentconfiguration in the past
one yeah that's exactlyright well of
course not becausein classicalphysics
the position of a particleand so forth
is a real numbera realnumber is
something that you can never determine
exactlyah andso there's always
imprecision andthat imprecision always
represents a degree ofuncertainty and
where you willbe nextnow it gets worse
andworse ingeneral it's likelyto get
worse andworse as you try to take
larger and larger time intervalsa given
degree of imprecisionin what you
actually know can get magnifiedand get
the magnifiedinto worse and worse
imprecision as time goes onso are in
practice in practiceclassical systems
don't really have the property thatyou
can predict endlessly where they're
going tobe and exactlywhat they're
goingto do butyou canalways saygiven
a time intervalI want to be ableto
predict exactlyfor the next 30 seconds
where every molecule inthis room will
be let's forget quantum mechanics now I
wantto be ableto predictexactly where
everymile
kyoool there will be then there's a
certaindegree ofprecision in the
present informationat exactly one
instantof time which will permit meto
be ableto extrapolate for 30 seconds
okay it willnot permit meto
extrapolate for 40 seconds if I try to
extrapolate for 40 secondsI willfind
the errors get magnifiedout of control
if I want to predictcorrectly for 40
secondsI will have to doeven betterin
my initialconditions and my knowledge
of exactlywhat thestate of the system
is so in practice this idea of
determinismis defectiveit's defective
because in orderto determine fora
given length of time you'll haveto have
precisionwhich is so goodthat it's way
way beyond anything anybodycan do
but in principlegiven any length of
time in classicalphysics thereexist a
degree ofprecision which allows you to
extrapolate for that lengthof timedoes
that answer the question
yeahah so some systems arevery
predictable some systems are less
predictable and get outof control very
quicklythey're called chaotic systems
but the principles arethe same that
it's just a degree ofprecision which
youneed to know in thebeginning in
orderto extrapolate for a given length
of time well youcould decide I'mnot
sure what thedifference between
predictableand detail the way I use the
terms I use them interchangeablyso I
can't say that
okay okayyeah allright what's that
yeahright the equations are
deterministic if you know theinitial
conditions are deterministic are
predictable
infinitely predictable if you know the
initial conditionswith infinite
precisionyou never doand therefore
they're never completely predictable now
the fact that youneed to know both the
position and the velocity inclassical
theory in the physics inorder to
predict whathappens nextreflects
itself inthe structureof the equations
of mechanics specifically it tells you
thatthe equations of motionNewton's
equations in this case are what are
called second-order equationsinstead of
first-order equations let me illustrate
itby startingwith a first-order
equationa first-order equation what a
first-order equation meansis thatit
only has quantities of first derivatives
with respect totime in other words only
contains velocitiessecond-order means
it containsnot only first derivatives
but second derivativesmeans itcontains
accelerations the equations ofmotion
acceleration is the second derivative of
the motionum we could write we know
what the realequations of motionof
Newton are F equals MA it contains
acceleration let's write a phony
equation let's write F equals mass times
velocity force isF velocity is velocity
and let'ssuppose thatforce just
depends on where you arewe have a
particlethat movesalong a line it's
subject tofour
whichvary alongthe line force may be
big here small there and soforth so the
force depends onposition and let's
imagine thisfake equation of motion
that it's equal tomass timesvelocity
andwhat is velocity velocity is the
timederivative of theposition the X by
DT and I will use continuously
throughout this course the notationthat
time derivative isjust indicated by a
dotthat means time derivative all right
what does this tellme what do Ineed to
know in orderto predict what happens
next I sayfor thisequation we only
need to knowthe position of a particle
if we know the positionof a particleI
can tell you what the velocityis just
from the equationif I knowthat the
positionis a particular position then I
know the force on it
and from the equation that tellsme what
the velocity isdoes it tell methe
acceleration how about the acceleration
let's see if I cancompute the
acceleration to compute the acceleration
fromthis equationwe just differentiate
it once morewe writethat the F by DT
is equalto mass massis just a constant
times the second derivativeof the
position orjust the accelerationso we
haveover here acceleration what about
the F DT the time derivative of the
forcethe force varies with time because
the position varies with time so the F
by DT is justa reflection of the fact
that the position ofthe particle varies
with time and we can write theF by DT
using standard rules of calculus as just
you did the change in F with respect to
position timesa changein positionwith
respect totime in otherwords the
velocityalright we've already figured
out what the velocity is knowingthe
force knowingthe position so we know
thevelocity
and we can readoff fromthis equation
what the acceleration is
we can figure outall ofthe derivatives
of the motion if we know where the
particle is by multiply differentiating
this equationso what ittells usto
make a short story out ofit is it tells
usthat if we know where theposition of
the particle at any instant oftime then
we know where it's goingto be in the
next instant thenext two instance the
next threeinstants it completely
predicts the motion butthis is notthe
character of Newton's equations Newton's
equations say F isequal tomass times
acceleration masstimes velocityso
let's lookat this equation F equals
mass times accelerationcan I predict
from this what the velocity isno this
is noequation for the velocity right if
I know what theposition isI know what
the forceis that tells me whatthe
acceleration isbut there's nothing in
these equations which tell me what the
velocity isthat means I have to addin
the velocity as a piece of information
to begin with I have no choice I have to
tell youin order topredict I have to
tell you the position as well asthe
velocity then if I know the positionand
the velocityI canthen predict the
accelerationthe next to a third
derivative the fourthderivative and all
of them so thattells me thatin order
to know where I amand where I'mgoing
to be I haveto knowthe positionand
the velocitythe phase spaceis atwo
dimensional spaceso we see then that
that classical mechanics does have this
character of aconfigurate or a phase
spaceof different configurationswith a
set oflittle arrows which tell you
whereto go next
but the phase space itself has a
position component to it and avelocity
component to it we willgo on and study
the classical equations of motionand
study themin a varietyof different
formulations but always the
connecting link willalways be
conservationof information the idea
that the laws ofphysics are completely
deterministicand described byequations
which tellyou where you will benext
that's thecharacter of classical
physicsokay bifurcatedit's one system
but you need two pieces of informations
that arewarn you I'm notsure what you
mean by bifurcated up here here you only
needknow now here youneed one piece of
informationto say where you'll benext
if youhere you stay here if you're
there you stay there yeah thestuff all
there is is heads and tailsthat's all
there isand if you knowwhere youare
you know where you'llbe next
no no we're in oh no no no no this one
rule
know this thisis one rule heads goes
two heads tails goes two tails that's a
singlerule and if you know that you're
at headsthen you know you'll benext at
heads if you know you're at tails
you know you but so you onlyneed the
piece of information heads or tailsthat
does itthat tellsyou everythingnow we
I suggested a differentdoes well yeah
you don't needto know where you were
before you only need toknow where you
arenow to know where you'll benext so
both of these laws require only
knowledge of where you areat one
instant to tell you where you'll benext
you simplyfollow the if you start
someplaceyou follow the arrowuntil you
comeback or youfollow the arrow till
you getto the next place and thattells
you where you'llbe nextyou don'tneed
anymore information than that the only
question is what one of thesepoints
corresponds todoes oneof these cut
points correspond to howmuch
information does it correspond to a
point inthis space
is it enough toknow heads or tails to
know what you'll do nextor mightyou
need to know the two previous things
that's a different that'sa different
set up weighted stay with the heads and
tails for a minute let's supposethat in
order tosay what happens nextyou need
to know the previoustwo configurations
let's do let'swork out that example and
let's write it in this form let's make
up a law I thinkI hadone down before
heads heads goes - heads heads tails
goes to watch tails tails heads goesto
heads and tails tailsgoes to tailsokay
is there something wrongwith that oh
this onlydepends onEd's coffee right
you're right you'reright rightsorry
sorry sorry sorryheads yeah we want
this oneto go to tails uh second one
goes to headsgood yeah what what what
what say itagain it's also true yeah
yeah yeahyeah it's italso hard towork
an example isn'tit let's see hmm
switch the second one this one make it
tails okayI think that's perfectly all
right
you loseinformation why well okay
here's a here's aset up here's a set up
if you know theprevious two then you
know what happens nextin fact let's
let's now see if we can make a table out
of this nota tablebut a set ofpoints
the set of points then thereare four
possibilities heads heads heads tails
tails headsand tails tailsokay let's
seehas heads heads heads tailstails
headsand tails tailsnow supposing you
go from heads heads to heads
this is heads heads headsthat means you
go from heads heads to heads heads right
you go from heads heads to heads heads
now supposing yougo from let's see what
happens if youstart with heads tails
where do you go to soyou go from heads
head tails - tails tails okayokay what
happens if yougo from tails tails are
we going torun to troublearen't we
tails tails goes to tails so tails tails
goes to tails tailsbad notallowable by
the laws ofphysics and now tails heads
goes where tailstails goes to tails
tailsgreat tails heads tails heads goes
to tickles- heads tails so that comes
here well it seems to mewe go we should
beable to make a consistent law whoa
whatuh this one this one
Oh tails tails goesto tails tails goes
to heads yeah
okay so then tails tails goes to heads
and that means that tails tails headsso
I'm his tails tails goes toTed - tails
headsah oh okay good good now we have a
workable lawnow we have a workable law
the onlything we hadto do was to say
that what we originallycalled a
configurationnamely a single heads or
tailswas not a complete specification
informationa completespecification of
information involved
twopieces ofinformation and once we
recognized thatwe were able to to to
write thisdown as a law ofphysics
which is deterministic and reversible
okay so you don'tknow offhand you don't
know tobegin with what piecesof
information you need toknow in orderto
know whatto do next butthat's what a
stead that's what theconfiguration
space or that'swhat the phase spaceis
it's the collectionof all the things
you needto know to know what happens
next
now ofcourse youcould go beyondthis
you could say I need toknow the first
three things inorder to know what
happens next you can do thatyou'll
simply needmore states tomake a
deterministicsystem what wouldit mean
in classical mechanics to need more
information than the positions and
velocities suppose youneed the
positions velocities and accelerations
that were in third orderdifferential
equations but what would itsay about
thephase space it would mean youwould
need positions velocities and
accelerationsto represent the phase
space as it happensthat is not the case
for classical mechanics that's an
experimental fact but itwouldn't stop
us if we didif we did need the
accelerations we would justwrite third
order equations and wewill makeour
phase spacethree-dimensional
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