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review first of all the idea of initial
conditions and why we need both position
and velocity let's just take the case of
one dimensional motion as an example and
let me begin with a false law of physics
I think I called it last time
Aristotle's law Aristotle thought that
in order to keep an object moving you
had to continuously exert a force on it
of course we know that's not true an
object moving through free space under
its own inertia will keep moving without
an external force but Aristotle's law is
a consistent mathematical law let me
write at them what it says I don't know
that Aristotle of course ever said it
this way I'll say it this way I would
call up my own law except I know that
it's wrong so there's not much
percentage in it force is equal let's
put a constant binning call it mass
times velocity and we can make this a
vector equation or we can simply make it
the one-dimensional equation along an
axis it says that without force you
can't have any velocity if you remove
the force an object will instantly come
to rest a wrong equation but we can also
write this of course f equals mass times
dx/dt and just to remind you I often use
the notation dot on top of a symbol to
indicate derivative with respect to time
so that's Aristotle's law now what do
you need oh and we'll also assume
something else will assume that the
force law is a law which tells you what
the force on the object is if you know
where it is where the object is so let's
write f of X f of X is equal to mass
well it's mass times the X DT
now I say that in order to follow the
system all you need to know is at the
end at the initial starting point all
you need to know is the position you
don't need to know the velocity how do I
know that well look at this equation
supposing you knew that at time T equals
zero this is X the vertical axis doesn't
mean anything right now supposing we
knew the particle was at X then we would
instantly know the force Armit Y because
the force is a known function of
position but from that we would
instantly read off the velocity just
from the equation itself
so with Aristotle's equation if we know
X at an instant we also know the
velocity now what about the acceleration
do we know the acceleration let's prove
that we can construe that we can figure
out the acceleration to figure out the
acceleration all we have to do is
differentiate this with respect to time
will assume the mass in fact from here
on in forever and ever mass is a
constant that doesn't change with time
the mass of an object just stated of
course the mass of an object could
change we could drop the bottom law
office and its mass changes but that's
only because we ignore it part of the
system that fell off alright so here we
are F equals M DX DT let's differentiate
this with respect to time then on the
right hand side will of course have the
acceleration mass times the acceleration
what about the left hand side the left
hand side will have the time derivative
of the force so let's come over to this
blackboard and calculate the time
derivative of the force why does the
force change in the first place it
changes because the particle moves and
depending on where the particle is the
force will change all right this force
only changes because the particle moves
so let's calculate the time rate of
change of a force
that's equal to the top of the to the
derivative of the force let's call it
the derivative of the force with respect
to position I'm using some crazy mixed
notation partial and the derivative of
the force with respect to position that
tells you how much the force changes
when you move from one position to
another times how fast you move the X DT
in fact if you look at this equation
you'll notice it's correct because you
can cancel out the DX the F DT is the F
by the X DX by DT and what it tells you
is the force changes with time because
it varies with position and position is
varying with Bo with respect to time so
this then is the F by DX times the
velocity okay let's put it in here the F
by DT is equal to the F by DX times the
velocity let's forget this for a moment
well that immediately now tells us the
acceleration we know the force is a
function of position we can
differentiate it and find out how it
varies with respect to position so if we
know f of X we also know DF by DX and
the first equation up here tells us what
the velocity is if we know the position
if we know the instantaneous position we
know the velocity if we know the
velocity then we know the acceleration
so without any further information using
the equation we can also deduce the
acceleration of the object we can go
further what's the derivative of
acceleration called jerk jerk the time
derivative of acceleration let's say we
can work that out just for fun just
supposing we differentiate this equation
with respect to time then we'll have M
times the jerk the derivative of the
acceleration with respect to time let's
go skort jerk J which is derivative of
acceleration with respect to time all
right now we have to differentiate the
right hand side the right hand side is a
product to differentiate a product we
use the product rule and that says first
of all we have DF by DX times DV by DT
but that's just the acceleration that's
one term that's the term coming from
differentiating velocity and then the
other term is gotten by differentiating
the F DX we're differentiating with
respect to time but that's just the
second derivative of F with respect to x
squared times the velocity forget this
for a moment the time derivative of F of
the derivative of F is the second
derivative of F with respect to x times
the velocity again that's exactly the
same thing as over here except applied
to the derivative of F now we have to
multiply that by another velocity
derivative of F with respect to x times
the velocity times another velocity the
OP shot is in fact we do know the jerk
we a moment ago convinced ourselves that
if we know the initial position we also
know about the velocity in the
acceleration well all that goes in here
is velocity and acceleration and
knowledge of the force law so you also
know the jerk I don't know what the term
for the net the derivative of the jerk
is the putz the schmuck I don't know but
I don't know what it's called but you
can see the pattern differentiate it
again and each time you can deduce the
next derivative that in effect says you
can build up the entire motion if all
you know is the
at one instant of time now as I said
this is a false law so let's go to
Newton's laws and ask the same kind of
question so f is now equal to mass times
acceleration which means mass times the
second derivative of position with
respect to time sometimes if I use my
dot notation I'll call that M X double
dot each dot means the derivative with
respect to time okay let's suppose again
I know where the particle is at time T
equals zero that means I know X that
also means I know the force on it
because the force is a function of
position but how on earth am I going to
get the velocity from this I can't get
the velocity what it tells me is the
acceleration in terms of the force and
there is nothing in this equation which
tells me what the velocity is just
doesn't tell me so I've got to add in
the velocity as another piece of
information X and velocity or X and X
dot or what I need to know to get
started well if I know X and the
velocity then I can calculate the jerk
well first of all I can calculate the
acceleration the acceleration I
calculate from Newton's equation so now
what do I know I know the position I
know the velocity because I'm assuming
that was given to us and I now deduce
the acceleration from there on I can
continue and compute the jerk in exactly
the same way if I know the the
acceleration and the velocity I then
compute the jerk and then the multiple
all the multiple derivatives so the
lesson is in Newton's equations which
are second-order equations which in
all second derivatives with respect to
time you need to know to get started
positions and velocities you need to
know the position and velocity of every
particle and every component of the
position and velocity if you know that
then Newton's equations allow you to
compute and of course you also need to
know the force law as a function of
those positions the force law might
depend not only on the position of
particles but it might depend on the
different other position relative
positions and so forth but if you know
all the positions then it's assumed that
you know the force all the positions of
all particles then it's assumed that you
know the force on every single one of
them due to all the others that's the
assumption of Newton's equations now we
also talked about conservation of energy
I think I was getting a little bit tired
I tend to do that after two hours so
let's just go back quickly and discuss
conservation of energy from Newton's
equations nothing deep at this point I
mean conservation of energy is clearly a
lot deeper than Newton's equations it
follows for electromagnetic radiation it
follows in the theory of electrodynamics
it works in relativity theory it works
in general relativity it works in
quantum mechanics there's clearly
something much deeper about the
conservation of energy than just what
we're about to demonstrate but still
we'll start slow ok now I gave you the
example a couple of examples where
energy conservation fails for example if
this friction force is of course energy
conservation really does work it works
if you were to follow the heat and
everything else that's produced by the
sliding eraser but if you ignore all
that and just think of friction forces
slowing the eraser down it would seem
that the energy the kinetic energy the
energy of motion suddenly goes away and
not it's not compensated for any change
in potential energy so there are
obviously formulas that you can write
down which would not conserve energy
all right give you another example the
other example was the spiral staircase
the spiral staircase you can think of
this way supposing you have a force law
that always pushes you around a circle
in the same direction the force always
points around the circle let's not worry
for the moment where the centrifugal
force takes you off the circle let's
pretend that there's a good solid
barrier another force which keeps the
particle from moving off the circle and
what's going to happen if the force is
always pushing you around the circle in
the same direction the particle is going
to get faster and faster and faster and
it's clear will by the time it comes
back to the same position it will be
going faster there's no possibility of a
of a conservation of potential and
kinetic energy you've come back to exact
exactly the same spot so therefore the
potential energy if there is any will be
the same but the kinetic energy has
increased and it'll keep increasing a
force law like this is again unphysical
it never occurs in nature the forces
that occur in nature are called
conservative and they are always of the
following form the components of a force
I'm speaking about a single particle now
we'll generalize this to many particles
in fact we'll generalize it to arbitrary
systems but let's take the case of a
single particle the force and now think
of the force as a vector I'm drawing it
in two dimensions it could be in any
number of dimensions is equal to minus
the derivative I'll right click I'll
write a formula and then I'll explain
what it means this formula this triangle
is simply a symbol for derivatives
partial derivatives
imagine that the potential energy is a
function of position and if the position
were two-dimensional as it is here there
would be a function of x and y this is
the potential energy you are this
formula really just means that each
component of the force with the X
component of the force and the Y
component of the force are equal to
minus the derivative in this case with
respect to X of U of x and y and minus
the derivative with respect to Y of U of
x and y that's what the symbol means it
means the collection of derivatives
partial derivatives with respect to the
individual coordinates so what it says
then is the components of force in a
given direction are proportional to how
fast the potential energy is changing
along that direction
alright let's use this to prove the
conservation of energy for the motion of
a particle I did it last time but I as I
said I was getting a little bit tired so
I want to do it again and just do what
its review okay let's define the
quantity of the total energy as the sum
of kinetic energy plus potential energy
okay what about what is kinetic energy
I'll define it for you now but
definition for a particle moving its
one-half the mass of the particle times
the velocity squared the velocity
squared means if there's more than one
component of velocity
it means the sums of the squares of the
components of the velocity so in this
case with two dimensions it would mean
one-half m VX squared plus V Y squared
more generally I could write it as
one-half the mass of the object times
the sum over the components of V I
squared
some meaning the x squared plus y
squared plus Z squared or whatever the
number of directions of space is okay
that's the kinetic energy so that's
usually called T again just like P
stands for momentum P stands for kinetic
energy don't ask me why and then the
other contribution to the energy yeah
it's also our you as potential a lot of
people use v for potential but that gets
mixed up with voltage which is potential
energy divided by by charge so I use you
and I steer clear of V but other people
use V it also gets confused with
velocity T Plus u that is the total
energy I want to prove that it's
conserved as a consequence of Newton's
equation F equals MA what does that mean
that it's conserved it means that its
time derivative is zero so all we have
to do is take its time derivative use
Newton's equations and prove that
everything cancels out when we take the
time derivative that's the exercise and
it's not very hard let's do it
okay so let's calculate the time
derivative of the kinetic energy
derivative with respect to time of
kinetic energy that is here it is right
here here is the kinetic energy one-half
MV squared M is not changing mass is
fixed one-half is not changing the thing
which might change is the velocity all
right this of course is the sum of
squares of velocities so it's a sum M V
sub I times DV sub I by DT now what I've
done as I've said that the derivative of
a square namely the derivative of V
squared is V
time's the derivative of B the
derivative of the square of something is
the something times its own derivative
whoops I forgot a factor I did it right
but I think it didn't say it right the
derivative of the square is twice that
something times the derivative of
something all right that gets rid of the
factor of two so the time derivative of
the kinetic energy is the sum over all
the directions of space the mass times
the velocity times now what's the what's
the V sub I by DT that's the component
of the acceleration so we might as well
call it the acceleration a sub I I the
component of acceleration let's
calculate the time derivative now of the
potential energy and the potential
energy is also is a function of position
when I write here that it's a function
of X it could be a function of X Y Z all
of the coordinates what is the
derivative of U with respect to time
well u varies with time for the same
reason that it did before because the
position varies it varies a little bit
if you move in the X direction it varies
a little bit if you move in the Y
direction it varies a little bit if you
move in the Z direction
and so in fact the derivative of U with
respect to T is a sum of terms the do
each term being the derivative of u with
respect to x times the time derivative
of X plus D u by dy times the time
derivative with respect to Y and so
forth or I can summarize it by saying
it's the derivative of u with respect to
the ithe coordinate x times the velocity
in the eiection time
component of the velocity in the I
direction again summed over I so now
let's write down the total time
derivative of the energy it's the sum of
these two derivative of kinetic energy
derivative of potential energy and we
can write down that the time derivative
of energy is equal to the summation of
all the directions
let's take V sub I first and then write
ma sub I that's the first term here you
can see where this is going and grouping
em with a sub I for a reason okay and
then we have to add to that again a sum
over I and it is plus bu by the X I also
times V I both these terms have the
velocities in them so let's take the
velocity outside the parentheses like so
well what do we have
what's ma sub I is a force mass times
acceleration so this is the component of
the force in the eiection but what's the
U by DXi that's minus the force in the
eiection is minus sign here
so this is just F minus F that's what's
in the bracket here notice the force is
minus the derivative of the potential
energy all times V but left F minus F is
zero whether it's multiplied by V or not
so term by term all you get a zero
upshot energy is conserved and now we're
only talking about a single particle
moving in a force field but we could be
talking about any number of particles
instead of thinking of the sum as just
to sum over the three directions of
space it could also include the sum over
different particles and precisely the
same kind of analysis would say that if
you have a system of particles that the
system of particles will have a
conserved energy no matter what the
force is as long as the rule is that the
force on every particle is minus the
derivative of U with respect to its
position okay so that was review I don't
remember if we talked about conservation
of momentum last time we know okay so I
will now begin reviewing what I did not
do last time but which I had intended to
do and so let's review conservation of
momentum now as I said this was a little
bit of trickery and a little bit of
magic and you might say yeah but why is
energy really conserved we will come to
that but for the moment we're just
observing that Newton's equations allow
you to define a certain thing namely T
plus V to be called energy okay
now let's come to momentum now first of
all let me rewrite Newton's equation in
a slightly different form nothing very
deep no talking about a single particle
the single particle is just has the
equation the force again we can do it in
components F sub I is just equal to mass
times the derivative of V sub I with
respect to T again I simply stands for
one of the three coordinates of space so
these are the three coordinates alright
mass doesn't change with time so I can
bring it into the time derivative and
just write the force I'm not going to
bother putting indices down here is the
derivative with respect to time of mass
times V and now give mass times V a name
the name is momentum okay
mass times velocity and it's a vector it
has components it has we can write this
for each component are M times V is
called momentum and so another form of
Newton's equations is that force is
equal to the time derivative of the
momentum of an object momentum is called
P and so we have we can make this a
vector equation if we like force is
simply the time derivative of the
momentum I don't know
I don't think Newton actually defined
the definition of momentum as far as I
know I'm not sure it's been quite a
number of years since I read the
Principia
what assumptions are we making about
time well no we didn't talk about it
well we're assuming that there's a
universal concept of time at every point
in this room we could have a watch or a
clock each one of the other clock
they're synchronized and we're making
the dumbest simplest the Newtonian
assumption that time is an absolutely
universal thing at every point of space
we can synchronize our watches so that
everybody will agree upon the time and
that time just flows uniformly in in the
most naive kind of sense one of the
things we have assumed is that the force
law here does not explicitly depend on
time we could imagine force laws which
change with time I'm assuming for the
moment that that's not the case all
right now let's come to conservation of
momentum first of all even if there was
just a single particle and there was no
force on it momentum would be conserved
because who would say the time
derivative the momentum is equal to zero
so for a single particle with no forces
on it out in outer space far away in
between galaxies the momentum is
conserved at least to the approximation
that we ignore forces on a particle but
that's that's hardly a very general
example so let's let's think about a
more general example the basic
foundation for the conservation of
momentum is another one of Newton's laws
of our force here's what Newton said he
said the forces between objects well
first of all he said the force on every
object and by every object we can mean
particles now everything being made out
of particles the force on every object
is the sum of forces due to all the
other objects all right so if we call
this particle one
then the force on particle 1 is equal to
the force on particle 1 due to particle
2
let's call that F 1/2 plus the force on
particle 1 due to particle 3 we assumed
that a particle doesn't exert a force on
itself plus f1 4 and so forth force on
particle 1 is the sum plus 1 2 3 4 5
same thing for particle to the force on
particle 2 is the force on to do 2 1 ok
so notice what I've noticed my notation
my notation is the first entry is the
particle that we're talking about who's
we're talking about the force on the
first particle here the second entry is
refers to the particle that the force is
due to so this is the force on 1 due to
2 here is the force on to do 2 1 plus
the force on 2 2 2 3 plus the force on
to do 2 4 and so forth let's just take 3
PI I don't want to write too much let's
just truncate it at 3 particles and
finally the third particle would have a
force due to a force on 3 due to 1 plus
a force on 3 2 to 2 and that's all
now Newton's assumption Newton's law
it's called action and reaction of
course is that the forces of a particle
on another particle the forces between a
pair of particles are equal and opposite
so the force on 1 due to 2 might be an
attraction for example is
equal and opposite to the force on to do
two one so if two is pulling one upward
to the left one is pulling two downward
to the right with the same magnitude of
force now there was another element to
this that the force is along the line
separating the two particles but we're
going to not worry about that right now
that will come back to that the
important point is that the force on one
due to two is minus the force on to do
to one we can make a vector equations
out of these if we like they are vector
equations and what did the equation what
does Newton's law say it says that F 1 2
is minus F 2 1 F 1 3 is minus F 3 1 and
F 3 2 is minus F 2 3 ok now let's see if
we can explore momentum conservation
what we'd like to prove is that the time
derivative of the total momentum of a
system is zero as a consequence of this
what do I mean by the total momentum I
mean the total sum of momentum on all
the parts of the system all the
particles that make it up
so let's write them down let's do the
three particle case the P 1 by DT I'm
not going to bother putting vector signs
above the above the think vector the P 1
by DT is equal to F 1 which means F 1 2
plus F 1 3 plus all the others if there
were more what about the P 2 by DT
that's equal to F 2 1 plus F 2 3 and
finally DP three by DT is equal to F 3 1
plus F 3 2 now let's add them up the P 1
by DT DP 2 by DT and D P 3 by
he that gives us the total time
derivative of the total momentum so the
time derivative of the total momentum DP
by DT is just the sum of everything in
the right hand side but they cancel in
pairs f12 is minus f2 1 this is just
saying the forces cancel f1 3 as minus
f3 1 and F 2 3 is minus f2 3 they all
add up to zero and it says that the
total momentum retired total momentum of
any closed system means a system with no
external forces on it of course we might
have other forces due to some other
object in the system then we would have
to take it into account also but if we
have a closed system a closed system
means we've accounted for everything in
the system that can exert forces then
the time rate of change of momentum is
zero and momentum is conserved again it
comes from a rather special assumption
about Newton for Newtonian forces but
it's far more general it applies to
light if a light ray bangs into the wall
it'll give the wall a little bit of push
it'll change the momentum of the wall
why because the light ray has some
momentum it applies in quantum mechanics
it applies in the general theory of
relativity so it's far far more general
than this little example shows and our
object is to will eventually be to write
the laws of physics particular Mechanics
in a form which is so general that we
can see why these things had to happen
okay any questions yes
loud in Newtonian physics it is alright
in classical mechanics Newtonian physics
and Newtonian physics means forget
relativity in Newtonian foot it this way
a court in Newtonian physics the speed
of light is infinite you turn on the
lights and the rooms lights up
instantaneously there's no lag in the
real world of course forces do not act
instantaneously it takes time for a
force when you when you do something to
a particle over here it takes time for
the force to respond over there
that's because of relativity special
relativity we're assuming lutonium
physics and lutonium physics means that
forces are instantaneous so yes we're
assuming they're instantaneous again ah
what's curious is that special
relativity undermines that
instantaneousness of forces but it does
not undermine the conservation of
momentum so it's somehow much deeper
then than our examples indicate it's
easier to derive right there are more
momentum was in there is energy law how
many momentum laws are there for a
single particle for example or how many
momentum laws are there altogether for a
combined four point system three one for
each direction of space how many energy
laws one so altogether there are four
conservation laws three for three for
the three directions of space and one
for the direction of time and in the
special theory of relativity they form a
complex of four related conservation
laws but this this what's more general
than that okay
now it's my usual yeah we're going to
need a little bit of mathematics not
heavy-duty mathematics fairly light but
let's spend a little bit of time we're
going to do next is we're going to go on
to what really is the deep law of
classical physics it's called the
principle a law of least action and it
contains all this junk the law of least
action has all of this stuck into it
plus a lot more but before we do we need
some math elementary mathematics
extremely simple
Neph that you know very well I just want
to spell it out and remind some of you
who haven't done calculus for for a
couple of years so that so that we're
all on the same playing field
the first thing has to do with
minimizing functions this is very
elementary I'm not telling you anything
you don't know if you have a function of
several variables let's start with a
function of one variable function of Y
some known file is not force it doesn't
say F does not stand per force now it
just stands the function f of Y and Y is
just some generic independent variable
okay here it is here's the function
that's plotted vertically I plot F
horizontally I plot the independent
variable Y and it has some shape
something the first fact is if you want
to find the local minima the local
minima of the function mean the places
where if you're there if you go in
either direction the function increases
a global minimum would mean the absolute
minimum the lowest possible minimum but
we're going to speak about local minima
local minima mean that if you are
confined to that region you would think
that the function was minimum right at
that point what is the mathematical
condition for searching for minima
what the derivative of a function should
be equal to zero
right that's not sufficient
yeah yeah right
so the first condition and the easiest
one is that at a minimum the slope of
the function is zero that says that the
derivative of the function is equal to
zero at minima it's also zero at maxima
notice the first condition is that DF by
DX or DF by dy total derivative in this
case total derivative if it's only a
function of one variable is equal to
zero now there are other conditions
there are the other conditions are
basically there to sort out two other
cases the maximum the F by dy is also
zero at Maxima but you can check whether
it's a maximum or minimum by looking at
the second derivative the second
derivative is positive at a minimum and
negative at a maximum but we're not
going to worry about that distinction
the principle of what's the Jenna Jenna
ik term for a function which is either
maximum or minimum and you don't may not
know which extremum is one word
stationer in fact in fact it's even
another possibility an inflection point
where the function is flat and then goes
down like that again zero derivative at
that point X cubed is a function which
has a an inflection point like that at
the origin its derivative is also zero
at the origin knowing that the
derivative is zero doesn't sort out
these three cases of minimum maximum or
reflection point so the word for e the
minimum of max
is extremal extremum point but what's
the word for any one of these three
stationary stationary stationary the
function is stationary at those points
which means that if you move a little to
the left or move with a right it neither
increases nor decreases well it may
increase the second order but it doesn't
increase the first order the derivative
is zero there it's said to be stationary
the real principle of least action is
really the principle of stationary
action we'll call it the principle of
least action and always pretend that
we're looking for minima of things but
in fact it's more general the principle
of action is the principle of stationary
action but I'm not going to draw that
distinction after this point now the
condition for a stationary point is that
the derivative of F with respect to Y is
equal to zero and I'll often say minimum
but I will mean stationary point now
supposing we have a function which is a
function of several variables f of X and
y and let's think about it let's draw
the xy-plane at every point on this
plane the function takes on some value
we could describe the function by a sort
of contour map and while seeing contour
maps you know what they look like each
contour describing a particular value of
the function and as you move from one
point one contour point to another
contour point the function changes well
are we I don't need the contour map all
we need to do is to remember that the
function has some value at every point
on this two-dimensional plane now
supposing what I want is the minimum
local minimum with respect to two
variables what that means is a point
where the function is minimum no matter
what direction you move off in
it could be any number of dimensions I'm
interested in points where the function
achieves a minimum which means that any
direction that you move off in the
function increases however you move
think of it as the bottom of a bowl
right yeah well it was strictly speaking
with talking stationary but I'll use the
terminology of minimum and if you want
to think about it think about minimum it
you won't you won't be making any
serious mistakes because in fact almost
in almost every case in mechanics we
really are talking about the minimum
action yeah ah the forces are
conservative it's not a maximum but I
think that's probably not true if there
are velocity dependent forces like
magnetic forces although I've never
checked that so non perturbative I don't
know it's non conservative no no no no
no no magnetic forces are conservative
but they're different in nature then
let's not get into them at the moment
we'll come back to them I believe there
are cases in physics where the minimum
of the action is not minimum but they're
unusual so let's just say minimum
minimum minimum what's not there let's
not worry about that subtlety all right
well first of all let's suppose we are
here's a minimum right there and let's
look at the function whoops I meant to
go right through that point let's look
at the function as we move to the left
or right away from that point now the
function is only a function of one
variable think of it as a function of
one variable along this line it should
achieve a minimum right at this point so
that means as you move along this line
ignoring all the other lines the
function should be minimum at that point
and that says that the derivative of F
with respect to
X should be equal to zero but we can say
exactly the same thing about moving on
the vertical axis if the function is
minimum at that point it will be minimum
with respect to a vertical motion so
that says the function should have no
derivative with respect to Y if you have
a function of a whole bunch of variables
let's give them the generic name X sub I
now function of X sub I the condition
that you're at a local minimum is that
the derivative of F with with respect to
every one of the X sub i's every single
one should be equal to zero
let's do an example or two just to I
think that as a function of one variable
that we don't need to do an example
let's take a function of two variables
and find its local minimum just in fact
it will be its global minimum I'm just
arbitrarily making up a silly function
now all right here it is x squared plus
y squared plus 3x plus 2y let's put in
plus XY plus seven that's my function I
want to find out where that function is
minimum so I differentiate it with
respect to X and set the derivative
equal to zero that gives me one equation
do the same thing for y it gives me a
second equation
what are those equations for therefore x
and y they tell me where the minimum is
achieved so I have two equations for two
unknowns let's welcome out just as an
example are the derivative the partial
derivative of F with respect to X
the f by DX alright that's from this
term we just get Y from this term we get
plus 2x this term doesn't depend on X
altogether from this term we just get 3
and there are no other terms which
depend on X so this is the derivative of
f with respect to X pretending that Y
we're just a constant alright that's the
partial derivative and our rule is set
this equal to zero that's one equation
for x and y alright let's do the same
thing DF by dy
all right the F by dy is X plus 2 y plus
2 equals 0 here are two linear equations
for X and y you can sit down and solve
them notice that the 7 didn't come into
it at all why not because the 7 is just
the height or overall height of the
function and it doesn't change where the
minimum is alright so I'll tell you what
the answer is
it's X is equal to minus 8/3 and Y is
equal to minus 1/3 and if that's not
correct it's not important is it - it
there I lost the paper that I did it on
but it's it's not important
there is an answer to the problem as you
can see it's just two equations with two
unknowns if you have n unknowns in other
words if you have a function of n
variables then there are n equations
like this n equations for n unknowns
typically we'll have a solution not
always some functions don't have minima
for example if I took a function that
just was like that forever and ever it
wouldn't have a minimum there would just
be no solution there would be no point
at which the derivative vanished
sometimes it's more than one solution
for example if I took a function with a
few minima in it but
you'll have enough equations for enough
unknowns to locate solutions that's
that's mathematical point number one you
find stationary points or minima by
differentiating the function or another
way of saying it is finding points where
if you vary the function a little bit it
doesn't change it doesn't change the
first order in the variation basically
all right so that's that's a stationary
the principle of stationary functions
now we're not going to be interested in
minimizing functions of a finite number
of variables we're going to be
interested in something which at the
face of it seems far more complicated
its minimizing expressions which depend
not on some finite set of variables but
which themselves depend on whole
functions whole trajectories whole
trajectories minimizing quantities which
depend on a whole trajectory before we
before I give you an example let me just
say some words about what the basic
problem of mechanics is and more than
that the basic problem of classical
physics but let's say mechanics the
basic problem of mechanics is to
determine the trajectory of a system
from its initial starting initial
conditions that's the mathematical
problem that's now of course in every
case we may not really be interested in
determining if we have 10 to the 23rd
particles determining the trajectory of
that system is equivalent to determining
the trajectories of all 10 to the 23rd
particles nobody tries to do that
instead we mase make some statistical
the statements about it or some some
approximate statements but the very
precise question of classical mechanics
is determining the trajectory of a
system now what does it mean we spoke
last time a little bit about generalized
coordinate
generalized coordinates are simply the
coordinates of a system they could be
the Cartesian coordinates of the
positions of a system of particles they
could be angles they could be the
orientation of this eraser they could be
any number of different kinds of
coordinates that we might use for
example on the surface of the earth we
might use latitude and longitude to
locate an object those are all
generalized coordinates they and they
may not be Cartesian coordinates they
may be angles or other funny kinds of
variables but they do serve to locate an
object in space or locate a system in
space and those are called the
generalized coordinates and they're
often labeled by qq1 dot dot dot through
qn4 ordinaire for a single particle
there are three Q's XY and Z for two
particles there are six Q's at the XY
and Z of one particle and the x y&z of
the other particle if there are 10 to
the 23rd particles there are three times
10 to the 23rd Q's but as we've seen
already the Q's are not enough to be
able to predict the future if you want
to predict the future you also need the
velocities so in a more general abstract
language it also means that you need the
q dots the time derivatives you need 6n
pieces of information for n particles to
predict the future now what does it mean
to predict the future it means to
determine the trajectory of the system
which means q1 of T q2 of T q3 of T it
means to determine all of the
coordinates as a function of time all of
the particle coordinates or the angles
or the
orientation angles or whatever it is to
determine all of them not just one of
them as a function of time but all of
them such a system of functions as a
function of time we can call a
generalized trajectory a generalized
trajectory we can imagine horizontally
plotting all of the Q's
now horizontally I've just drawn two
dimensions here two dimensions will be a
stand-in for any number of dimensions q1
through QN and vertically we plot the
time a trajectory simply means exactly
what it says a trajectory through the
space and a trajectory through the space
is a Q as a system of Q's at every value
of time that's the mathematical problem
that that's put in front of us by the
questions of classical mechanics to what
extent can we determine the trajectory
from the starting point all right the
answer of course is given Newton's
equations we can predict the Q of T not
just from the starting point we need the
Q's and the Q dots to start with if we
know the Q's and the Q dots we can
determine the trajectory the equations
of motion the Newton's equations of
motion what they tell us is every point
along this trajectory they kind of tell
us if we're moving in a certain way how
to move next they tell us that if we
know the velocity along here or sorry if
we know where we are along here and we
know the forces then we know the
acceleration so that tells us if we know
where we are and we know where we're
moving we know where we'll be next and
we know where we'll be moving next the
way to say this is that Newton's
equations are local equations along the
trajectory and they tell us information
they tell us how to proceed from one
point to another the next along the
trajectory you don't need to know
anything about the rest of the
trajectory if you know where you are
over here and you know which way you're
going then you know where you'll be next
and you know where you'll be going next
so they're local equations and that's
why they're differential equations
differential equations or equations
among the positions and the derivatives
and so forth and that's the kind of
information that they have in them now
of course if you know this information
all along the trajectory then you can
build up the whole trajectory but the
equation itself is an equation which
tells you local information the
principle of least action is another
formulation of the same physics which
looks at the trajectory as a whole
instead of being given the position and
the velocity at a single time what
you're given is different kind of
information you're given the position at
the beginning of a trajectory okay it
could be the position of this eraser at
time T equals zero okay and after a
certain specified amount of time you're
given the position of the system at the
end of the trajectory in other words
you're given the two end points of the
trajectory so for example as an example
here's the eraser I tell you at time T
equals zero it's over here I'm not going
to tell you how it's moving so I can't
tell you what it's going to do next but
what I will tell you is that after one
second it will be over here there's only
one trajectory which goes from here to
here in one second there it is okay if I
throw it with the wrong velocity it
won't get there so it's just as good to
know two points along the trajectory as
to know the two pieces of information at
a single point along the trajectory same
amount of information instead of giving
the position and velocity at one point I
give you the positions at two separate
times that's the problem
that's another formulation of the
question of mechanics question of
mechanics instead of the initial data
problem the problem of finding
unique trajectory which connects two
points in this form the principle of
nature that governs trajectories is
called the principle of least action and
what it says is this some quantity
associated with this trajectory the
whole trajectory not just little pieces
of local information but the whole
trajectory than some quantity about that
trajectory which is minimum for the true
trajectory for the true trajectory and
will be larger for every deviation of
the trajectory from the true trajectory
true trajectory correct trajectory
here's an example take two points for
simplicity we could take them in two
dimensions calculate the distance
between them and ask what is the line
will let the distance would count draw a
line draw a curve between them throw any
curve between them calculate the length
of that curve and then ask what curve
minimizes the length what curve gives
you the smallest distance along the
curve between those two points of course
we all know the answer it's a straight
line okay but what's the mathematics
that goes into it the mathematics that
goes into it is minimizing a certain
quantity but that quantity is not a
function of a finite number of variables
that's a function of every point along
this curve so it's a much more
complicated object than calculating the
minimum of a function of a few variables
there's two approaches to it one is the
local approach which I'll show you what
it means in a moment and one is the
global approach yeah let's talk about
their local in the global approach the
global approach is just what I said you
pick your two points you calculate the
distance along the curve
of that's some quantity that depends on
the entire curve and then you ask what
minimizes the length along that curve
that's something it's a very global view
where you minimize something associated
with the entire curve now the local
point of view goes as follows all right
supposing we know let's focus let's take
our little microscope here and focus
along somewhere along the trajectory
over here in fact for our purposes let's
imagine breaking up the trajectory into
a lot of little pieces each of some
small length and let's focus right in
here the curve comes in over here so
it's pinned down over here it goes out
over here and I'm interested in where it
goes between them let's just suppose in
breaking it up into pieces as an
approximation as an approximation break
up the curve into little pieces and now
take two of those little pieces and ask
where should you locate the midpoint in
order that the curve be as short as
possible
now the answer doesn't depend where you
put this point does not depend on the
rest of the curve if you happen to know
this point in that point if you know the
point over here and the point over here
then the rest of the curve doesn't
matter you will have to in any case you
if you want to make the curve as short
as possible and you know that it passes
through here and here then all you have
to do is ask how do you make that little
segment between them as short as
possible what's the answer the answer is
obvious but it's a local version of the
same question it just depends on this
little region over here
the answer is obvious just go straight
ahead move this point until this little
piece of the curve as as short as
possible right it goes
ahead we could express this as a
differential equation we will as a
differential equation which basically
says that the difference in the slope
from one point to the next zero but it's
quite obvious that what it says within
this little red region here is just go
straight ahead if you generalize that
all along the curve that will tell you
at every point just go straight ahead so
this is the local version which says go
to each point on the curve and make sure
it's just going straight ahead that's
enough to tell you that the shortest
distance between here and here is a
straight line so there are these two
different versions of thinking about the
principle of least distance this would
be the principle of least distance
between these two points let's see just
for future reference let's work out
let's uh let's try to define our problem
mathematically even if we won't solve it
yet let's start define it mathematic
I've solved it for you where I told you
what the answer is a straight line of
course you knew that anyway but let's do
a little bit of mathematics and state
the problem in a mathematical form okay
let's let's take our curve and I think
I'd like to get a bit of pin
let's think about curve as existing in
the XY plane let's suppose we know that
it goes through this point that's the
point at x1 and another point at x2 and
let's think of Y as a function of X so
there's some curve and what we know is
the value of y at X 1 and we know the
value of y at X 2 in other words I tell
you where the end points are and I ask
you to find the shortest curve well the
first thing we're going to have to do is
calculate the distance along the curve
for any curve and then minimize it so
let's imagine let's not imagine let's
calculate the distance along this curve
we'll call this function y of X so if
you like this is a problem of minimizing
something with respect to a whole
function finding the function which
minimizes something or let's calculate
the distance along this curve if we know
Y of X first begin with a little segment
a little segment of size DX along the
x-axis all right so it's a little gap in
the function it's a little segment in
the function going over the interval DX
alright let's let's draw it over here
here it is it goes over a distance DX is
y of X and we can also speak of dy is dy
as we go from point X to X plus DX the
function goes from y to y plus dy what
is the length of the hypotenuse of the
triangle if we add up all these little
triangles and add up all the high
partner what's the plural of hypotenuse
I don't know
I guess so if we add up all the
hypotenuse is if we add up all the
hypotenuse is will then compute the
entire curve so let's compute it well
this side is dy this side is the X the
distance along here we'll call that DS
squared this is square of the distance S
stands for distance again one of these
letters that has nothing to do with with
the word S stands for distance the
square of the distance along this
hypotenuse here is clearly DX squared
plus dy squared or if I take the square
root it tells me the distance along the
hypotenuse let me write this in the
following way let me write d s I'm going
to take it over here i rewrite it over
here d s is equal to DX I'm factoring
out the x times the square root of 1
plus dy by DX squared where did I get
that from if I bring the DX inside the
square root I have to square it to bring
it inside the square root then the one
will become the x squared and this term
here will just become dy squared all
right so I'll just factor it out a DX I
want to think of X is the independent
variable B Y by DX that's just the
derivative of the function y of X so
that's nothing but the function the
derivative of the function if you if I
gave you a function Y you could
differentiate it and this would be dy by
DX all right now let's add up all of the
little intervals adding up all the
intervals means doing you need to grow
so the total length of the curve from
point X 1 to point X 2
caresse between one and two is equal to
the integral from x1 to x2 of just this
quantity here square root of 1 plus dy
by DX squared DX I'm not going to do
anything with this right now I just want
it to work out for you in terms of Y of
X if you know if somebody gives you the
function y of X this is the way you
calculate the total length along here
what is our problem
our problem is to find that function y
of X which makes this integral as small
as possible
now remember when I say that it's as
small as possible what's inside the
square root is bigger than 1 it's 1 plus
something positive you might say okay
look if I want to make this as small as
possible I will just make what's inside
the square ruler here as small as
possible and make dy/dx equal to zero no
that's not the answer to this problem
because we're making an assumption the
assumption is that the curve goes
between these two points
if dy by DX was 0 everywhere it would
just be flat it wouldn't go between
these two points so it's given that the
curve goes between these two points we
want to find the minimum value of this
integral the minimum value of this
integral and that means the curve Y of X
which does that for us
or the function y of X in this case if
we thought of X is kind of like time and
Y as the coordinate of a system this
might be determining a trajectory by
some funny principle of least distance
in space and time but for the moment
just all this is is how to find the
shortest distance between two points
now I haven't told you how to do it
we'll get to actually how you do it but
that's the mathematical problem find the
function which minimizes this
depression arm this s here depends on Y
of X you can sort of think of it as a
function of a function given Y of X
here's a rule for calculating a number
from it it's a function of a function
it's not a function of a few variables
it's a function of an entire other for
it's a it's a number which depends on an
entire function there's a name for a
number which depends on the entire
function it's called a functional a
functional is a function of a function
the result is a number but the input is
a whole function so the distance between
two points is a functional of Y of X and
our problem is to find the minimum value
of that functional subject that's to
some constraints that it passes through
oops that it passes through two points
let me give you another example couple
of other examples in physics where you
might want to minimize a functional with
respect to a function a principle of
optics is the principle of least time a
light ray goes from one point to another
point along a path that minimizes the
amount of time that it takes for the for
the light rate to go from that point to
another that's very much principle how
we figured it out I don't know but it is
a correct principle now you say that
just means the light goes in a straight
line if it minimizes the time that it
goes from one place to another and light
goes with the speed of light that must
mean it minimizes distance between the
two points yes that would be correct if
the speed of light was everywhere is
constant
but we're actually interested in a more
general and more interesting kind of
setup where you might have a material of
some sort with an index of refraction
which means the speed of light which
changes from point to point in the
material all right then it's not so
clear what trajectory to take in order
to minimize the time that it takes so
let me give you an example let's suppose
we have a material two dimensional
material there it is and the speed of
light varies for simplicity there's
nothing there's nothing special about
this but for simplicity let's suppose
the speed of light just varies with the
vertical axis here so we can divide this
up into little thin ribbons and each
ribbon the speed of light is different
for example the speed of light might
vary from some minimum to some large
value up at the top light goes slow down
here and incidentally it goes the same
vallah the same velocity no matter what
direction it's going in that's an
assumption there are materials where
light moves with different velocity in
different directions but I'm not doing
that
all right so if though if the light is
going this way it moves with a certain
velocity if it's going this way it moves
with the same velocity but the velocity
does vary from strip to strip okay now I
want to know what's the path so that the
time between this point and that point
is minimum that's the true trajectory of
the light ray well if you take a
straight line then that's not as
efficient there may be a path which
takes less time for example if I curve
the path up a little bit I didn't mean
to curve it quite that much but let's
curve it up that much anyway then it's
going to spend more of its path moving
where the speed is faster so it will
want to get the region's want to get
light doesn't want to do
anything but in order to minimize the
time that it takes to go from here to
here it might want to get to a region
quickly where the velocity is larger and
then make most of the path with a larger
velocity so it's not obvious that it
goes in a straight line and it doesn't
go in a straight line in fact it would
go in a trajectory something like that
if if the velocity of light increased as
you move vertically you might think well
maybe the best thing to do is to jump
straight up and then straight across but
no that's not as good because the upward
part of the journey doesn't get you
anywhere so it's not as good to just go
straight up and go straight across
there's some compromise where you curve
the trajectory to get it into regions
quickly where the velocity of light is
faster but not straight up and straight
across alright so what is then the the
quantity which you're trying to minimize
when you speak about the principle of
least time so I'll work it out for you
just for fun we'll work it out
again we have some solution which is Y
of X so we want to find the Y of X the
path which minimizes the time between
two known points and notice as a theme
here we're always talking about two
known points and asking for the
trajectory or the function which
interpolates between those two points
okay let's find let's see if we can
calculate the time that it takes to go
from one point to another first of all
again we'll break up the trajectory into
small little intervals all right each
interval has a distance which we've
already calculated here's the distance
all right so let's write that down again
over here the distance DS is equal to DX
times the square root of 1 plus the y by
DX the derivative squared all right but
on the other hand what is the speed of
light the speed of light is the distance
that it goes divided by
the time the time so each one of these
little segments will take a certain
amount of time right the rule is that
the S by DT is equal to the vasa T is
equal to the speed let's call that C but
C is not a constant it depends on Y I
could also take it to depend on X but
just for simplicity I'm imagining this
material which is a kind of layered
material with different speeds at
different heights so C is a function of
Y well that allows me now to determine
how long the light ray spins in each
little interval DX right we just solve
for DT so DT is equal to D s divided by
C of Y all I've done is is solved for DT
multiplied by DT and divided by C of Y
okay so that's how much time the light
ray takes to go from one point to a
neighboring point where they're
separated by distance D s but now here's
the s so we can also write this as DX
square root of 1 plus dy by DX squared
all divided by C of y
that's the time in terms of the function
y of X remember now what we're doing we
have some function y of X and we want to
see how long it takes light to go along
that function the Y by DX is just a
derivative if I gave you a function you
could differentiate it the X that's just
a little X interval from here to here
and C of Y is the velocity of light
along the curve at height Y what's the
total time that it takes to get from
here to here we add up all these little
line in all these little intervals of
time we add them up which obviously
means we do an integral and so the total
time between 1 and 2 let's call it t1 2
is equal to the integral DX from the
starting point x1 to the end point x2
start the end of here it is square root
of 1 plus dy by DX squared divided by C
of Y if I told you what C of Y was and I
gave you the function y of X we could
determine how long it takes the light
rate to go from here to here now that
doesn't solve our problem that just sets
up our problem it tells us for every
curve Y of X how long it takes to go
from point 1 to 2 then our problem is to
find the curve which minimizes the time
that's obviously harder all right right
so find the curve search to all possible
curves that you could search through and
find out what curve minimizes the time
well I'll give you another example some
other examples where for one reason or
another you might want to minimize
something with respect to an entire
function well here's a famous example
that everybody who builds bridges knows
about if you have a chain or a cable or
a string or whatever a weighted string a
weighted string or weighted cable hung
between two points now we're assuming
that we know the length of the chain the
length of the chain is fixed we might
ask what configuration does the chain
take and the answer always is it
minimizes the end of the potential
energy we would I will just tell you
that as a fact now it minimizes the
potential energy that means effectively
it gets as low as it possibly can
because the potential energy is just a
function of height potential energy
increases as you go up it tries to get
as low as it can right well it can't
drop down through the floor because the
chain has a fixed length R this sort of
configuration will not be terribly
efficient because we can gain some or we
can lower the potential energy in the
middle by letting it hang down deeper
here so there's some curve some function
of fixed length which minimizes the
potential energy I'm not going to tell
you what the function is I'm going to
remember what is it it's a catenary
right yeah it's catenary but if you ask
me what a catenary is I can't remember
so
that's another example of a problem in
which you might want to minimize a
certain function which depends on the
whole curve you might want to minimize
it with respect to the curve itself oh
it's just the height of each is
basically the height of each molecule it
doesn't matter because because you can
take it from the fixed points yeah but
if you were to change the zero of
potential energy we just add a constant
and it wouldn't change the the answer
but but right now I don't want to set up
I don't want to set up this problem all
I wanted to do is show you some examples
where physics requires that you minimize
something not with respect to a few
variables but basically with respect to
an infinite number of variables the
values of Y at every point X the
mathematics is called the calculus of
variations it's the problem of
minimizing a functional with respect to
a function or the principle of least
action is this kind of thing it is
minimizing something that is defined for
the whole trajectory now trajectory of a
mechanical system in other words a
system where that I drew write them that
I write them down here someone is young
okay not one function but a whole
collection of functions Q 1 of T Q 2 of
T remember what the Q's are they're the
generalized coordinates of a system Q 3
of T they're a whole collection of
functions which define a trajectory of a
system find the trajectory which means
find the collection of functions Q 1
through Q and of T which minimize
something
what is it minimizing what is the
quantity the quantity is called the
action
that's the principle of least action
that there exists a quantity called an
action which minimizing it with respect
to the trajectory is equivalent to the
usual rules of mechanics in particular
equivalent to Newton's laws of mechanics
but as in many many situations far more
efficient than far a far better tool
than just writing down Newton's
equations so there are equivalent
formulations of the same thing in which
one is easier and more useful depends on
the system but believe me there are
systems where the principle of least
action is a lot more efficient than in
particular if I were to take this eraser
which is composed of 10 to the 30th
particles and I had to write down all 10
to the 30th force laws between all three
times 10 to the 30th coordinates it
would take me a long time to write down
those equations if I tell you on the
other hand that the trajectory that this
object takes is moving through space and
rotating minimizes some action and that
I know how to write down the action it
is a much much more efficient tool for
many purposes yes yeah that's yeah that
the arm it implies Newton's laws in all
cases where Newton's laws would be the
underlying description or one under
language in any case where Newton's laws
apply it it is equivalent to Newton's
laws but it's much more general it's
more general for example it applies to
problems in special relativity to the
motion of objects and special relativity
it applies to fields it applies to the
electromagnetic field it applies to
basically every classical problem in
physics not just
motion of particles but motion of waves
fields the whole works the principle of
least action is the most general
formulation of mechanics or of classical
physics that we have so all systems in
classical physics that I can think of
have a formulation in terms of the
principle of least action now if it's
not particles but it's something more
complicated the principle of least
action might not you know we have to
generalize it but the little steps and
little steps the little step we're
taking now is to think about
trajectories for a system of particles
no no not yet yeah
right yeah yeah yeah um but there are
yes yes and of course that is an example
of using more general principles but
there are mechanical systems where you
wouldn't get too far with that either
and where the more general principle of
least action would be the most efficient
way to go about it all right so I will
tell you right now what the action is
all right now this action is going to
seem bizarre it's going to seem strange
it's going to be a little bit surprising
but the final acid test of whether it's
the right action is to test it out for
all Newtonian problems and see that the
principle of least action gives the same
rules as Newton's equations all right
once we do that we will then have a
reason to believe that this is the right
action so I'll tell you what it is it's
an integral over the trajectory so it's
it's now the independent variable is
time not X the independent variable is
time we have a whole bunch of Q's of T
or X's of T or whatever we like we
calculate the kinetic energy and the
potential energy and we integrate it
over time integral of the kinetic energy
which is what one-half for ordinary
mechanical system composed of ordinary
particles let's just think of it for one
particle well simplest thing one
particle in one dimension and then I'll
tell you what the generalization is one
particle moving up and down an axis all
right so the problem then is to find
X of T find the curve or find the
function X of T which minimizes a
certain thing which I will now tell you
what it is it's called the action you
might expect if I tell you that it has
to do with the energies you might expect
that it has to do with the kinetic
energy one-half MV squared which is
one-half dxdt squared plus the potential
energy integrated over time DT in other
words at every point along the
trajectory you might write down the the
action for a small interval of time
would be the total energy times DT now
this is wrong
bizarrely the answer is almost this but
not quite minus the kinetic energy minus
the potential energy that's called the
action the final in the final analysis
whether you call this plus something or
minus something is of course a matter of
definition you can you can redefine a
new variable and you can call it W which
is minus u and put in plus W here if you
like
but then the fine you can do that then
we have to find out what the solution of
the problem of minimizing the trajectory
with respect to this thing that you've
written down is what you'll find is
Newton's equations but you'll find
Newton's equations instead of finding
the force being minus W DW DX you'll
find it plus DW DX and you'll find out
that the thing that you call W is
behaving like the negative of the
potential energy so you can call it
anything you want I will call it minus U
of X here but this is the thing which is
called the action sometimes action is
called I sometimes it's called s I don't
want to use s because we've already used
s fo path
length and sometimes it's called a for
action this is the one case where I will
use a variable which is linguistically
or phonetically or whatever the word is
sensible what I suggest is we go through
a bunch of examples I'm throwing
something at you now and it is not easy
to explain where this comes from and why
this is the answer the best thing to do
I think is to examine it and find out
that it is the same as Newton's
equations and then we can go back and
say why did it have to be this to get
Newton's equations I rather not try to
motivate it at this point and just
examine it and see what we get
ok so let's let's hold in suspension our
questions about why this is the right
thing but let me notice some things
about it first of all let's compare it
with the principle of least distance or
the principle of least time the
principle of least time also had
integrals now in those cases are used X
as the independent variable and Y is the
dependent variable now I'm using time as
the independent variable and X as the
dependent variable so I pull the switch
on you of notation what I the
independent variable which is like X in
the previous example is now T instead of
trying to calculate Y of X I'm trying to
calculate X of T so the dependent
variable is X the independent variable
is T but let's just compare it with the
principle of least of least distance for
example or the principle of least time
the principle of least distance told us
compute one plus dy by DX squared
integrated DX if we wanted the principle
of least time the optical principle we
would also throw in here R 1 over C of
what of X I guess y sorry all right
first of all the thing which I'm
computing here depends on two things it
depends on Y of X and it depends on the
Y by the X so it depends on the function
and the derivative of the function with
respect to the independent variable
that's also true here this depends on
the function X of T and also on the
first derivative of the function with
respect to the independent variable so
it has the same kind of dependent
doesn't have a square root in it but
still the thing eeling in the integrand
here depends both on the first
derivative of the function and the
function itself depends on y of X and it
depends on dy by DX likewise here it
depends on X of T through the function U
and on the X by DT so it's a similar
kind of thing it's an integral along the
independent variable from one time to
another and we will also put in an
assumption which is very similar to what
we put in in the principle of least time
namely in the principle of least time we
said that the integral is between two
fixed points which we don't vary we hold
the two points fixed and we ask what's
the trajectory which takes the least
time between these likewise in the
principle of East at least action we
assume that we know what X at the
beginning of the trajectory is let's
call that t0
and X at the end of the trajectory t1
given not to be changed so it's a
trajectory between two points in space
and time if you like well you can see
what the end analogy is as I said the
basic idea is instead of giving the
position and velocity you're giving the
position at two distinct times find the
trajectory which minimizes this integral
of the kinetic energy minus the
potential energy now it's very
surprising it's even still surprising to
me after 40 years of having studied this
for more than 40 years and I know the
reason why it is I'm still always
surprised that it's the kinetic energy
minus the potential energy all right so
you should also be surprised that it's
the kinetic minus the potential but it
is that's the action the thing in the
integral here the integrand is called
the Lagrangian the kinetic energy minus
the potential energy is called the
Lagrangian and in a general form the
scripto it's a function of all of the
Q's it's a function of the Q of T's and
the Q dots of T's the T the action is
always an integral over time from the
beginning of the trajectory to the end
of the trajectory the integrand depends
on the coordinates X and the velocities
X dot or Q and Q dot that's the most
general version of the principle of
least action namely that there exists
something is the law of physics the law
of physics says that for every
mechanical system there exists a
quantity called the Lagrangian
which you build the action and then you
find the path or a trajectory which
minimizes that action in simple cases in
fact in all cases that correspond to
Newtonian physics of one kind or another
this Lagrangian is always the difference
between the kinetic energy and the
potential energy and more general
systems which don't have a Newtonian
analogue this Lagrangian may well be
something totally different than
peculiar but it will still depend on Q
and Q dot and and the principle of least
action will be more fundamental than any
kind of Newtonian laws of physics in
particular the this is a Newtonian rule
the motion of a particle in the special
theory of relativity is quite different
than this but it still has this form ok
so what we're going to have to do next
time we won't do it tonight is learn how
to find given a Lagrangian and given an
action like this how do we convert the
rule of the principle of least action
into equations of motion that look like
Newtonian equations of motion how do we
go back and forth from the Lagrangian to
Newton's equations of motion and if it's
not Newton's equations of motion what
are the equations of motion that follow
from a given Lagrangian notice that this
if we knew how to do this if we knew how
to solve such problems we would solve
among other problems by the same rules
the optical problem of finding the path
of least time so there's a close analogy
between optics and classical mechanics
principle of least action and mechanics
principle of least time and optics but
the methods the techniques are exactly
the same so the next time we'll work out
what are the equations of motion which
follow from a particular action
principle
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