touchy about first university please
stanford dot the
amira 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 wall physics
i think a cold last time profile for small
there are other stock all thought that more people object moving
you have to continuously exert a force on it
course we know that's not true
object moving from free space
uh... under its ownership will keep moving without an external force
but i was tunnels lol
is any consistent mathematical all of you write it down
where it says
uh... i don't know that our stop of course of the senate this way i'll say
it this way
are
i would call on my own walks up and know that the wrongs and numbers percentage
in it
forests
is equal let's put a constant denim coat map
bositis
and we connect this effective equation or we can simply make it that
one-dimensional equation along an axis
it says
that without force you can't have any loss every
if you remove the force
project was instantly come to rest d
wrong equation
but are all we can also write this of course
equals mass
primes the exp t
and just to remind you often use the notation
dot on top of a symbol to indicate
derivative with respect the time
so that's aristotle's loft
now what do you need
or it will also assume something else
will assume that that the force law
is a law which tells you what the force on the object is if you know where it
gets
where the object is
so let's write f of x
f of x
is equal to mass
mass times the exp t
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
we'll look at this equation supposing you knew
time equal zero missus x
vertical axis doesn't mean anything right now
supposing we knew the particle wasn't xd
we would instantly know
before sonic
white because the forces unknown 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 expand an instant
we also know the velocity
know about the acceleration
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 the time
will assume the mass and back from here on in forever and ever
mass as a constant it doesn't change with time
the mass of the market
just state of course the massive an object could change it to drop the
bottom or persons missed changes
but that's only because we'd nord part of the system along
are totally off
f equals and the x_ detail it's differentiate this with respect the time
that 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
we'll have the time derivative of the force
so let's come over to this blackboard
and calculate the prime derivative of the force
why is the force change in the first place
it changes because the particle moves
and depending on where the particle layers
the force will change
force only changes because the particle moves
so let's calculate the prime rate of change of a force
that's equal
for the time for that
through the
derivative of the force
might score riveted before us
with respect to position
amusing some crazy mix notation partial and uh...
derivative of the force with respect the position that tells you how much the
forced changes when you move from one position to another
times how fast you more
the exp tape
in fact if you look at this equation you'll notice it's correct because you
can cancel out the d_x_
the f_t_c_ is the f by the exp explaining t
but what else do is a force changes with time
because of their lease with position and position is bearing with the uh... with
respect to time
so this then is
the f_b_i_ d acts
hamesha velocity
okay let's put it in here
the f five eighty
is equal
the f_ by the acts
times the velocity
we get this for a moment
well that immediately now tells us the acceleration
we know the forces of once in a position
weakened differentiated and find out how it varies with respect to position
so if we know af of acts we also know the f by the x
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
without any
further information
using the equation we can also be deuce
the acceleration of the object
we can go for that what's the rhythm of exhilarated called
jerk
time the river acceleration of the work that out just for fun
supposing we differentiate this equation with respect to time
and will have and
primes the jerk
derivative of the acceleration with respect to converse chris korda jerk
jerry
derivative of acceleration with respect to prime
right now i 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 the f by d_x_
times devi by deeply
but that's just the acceleration
one-term
that's the term coming from differentiating velocity in the other
terran
is gotten by differentiating the f_b_i_ acts
were differentiating with respect to pardon
but that's just
the second derivative of f with respect to act squared
times the velocity
forget this for a moment
time there were a bit of a bath
the river a ref
is the second derivative of f with respect to x times of a loss of the
again
that's exactly the same thing is over here
except applied for the derivative ref
now we have to multiply that by another velocity
the review of applications at the x
velocity
barnesandnoble velocity
the outsiders
in fact we don't know the jerk
week a moment ago convinced ourselves that
if we know the initial position
we also know both the velocity in the acceleration
well all it goes in here
velocity
an acceleration
and knowledge of the four th law
you also know the jerk
i don't know what the term for the method derivative of the juror is
the pups marco dr
i don't know what it's called
but you can see the pattern
differentiated again
and each time you can deduce the next uh... derivative
but in effect says you can build up the entire motion
if or you know is the position at one instant of time
now as i said this is a full swap
so let's go to
there's laws
and ask the same kind of question
who's now we've got a mask arms 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 imap
x bubble dot each dot means of derivative with respect to time
okay let's suppose again i know where the particle is
at time t equal zero
that means i know x
that also means i know the force on it
because the forces of function of position
but how on earth and i going to get the velocity from this
i can't get the velocity what it tells me it is the acceleration in terms of
the force
and there is nothing in this equation which tells me what the velocities
just doesn't tell me
so i've got that it in the velocity
as another
piece of information
x
and velocity
or x
annex dr
a lot i need to know to get started
well if i know expendable ah city
then i can calculate the juror well first of all i can calculate the
acceleration
the acceleration
i calculate from the equation
so now i know i know the position
i know the velocity because sent that something that was given to us
and i know deduce
the acceleration
from their own
i can continue and compute the jerk
in exactly the same way
if i know the uh... the acceleration and the velocity
and in computer jerk
and then the uh...
uh... multiple the multiple records
so lesson is
and
in newton's equations which a second-order equations which involved
second derivatives with respect to time
you need to know to get started
positions and velocities
unique another position and velocity of every particle and every component of
the position and velocity
it's you know that
then newton's equations
allow you to compute
uh... and of course you also need to know the force law
as a function of those positions
before slow mike the pain
not only on the position of particles but it might depend on the different
than the present on the relative positions and so forth
but if you don't want positions
that it's assumed that you know the force
all positions wool particles
let's assume that you know the force arm every single one of them
due to all the others
that system should have been two questions
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
are so
let's just go back quickly and discussed conservation of energy
from newton's equations nothing deep at this point
marine conservation of energy is clearly a locket deeper than newton's equations
it follows
for electromagnetic radiation that follows in the theory of electrodynamics
it works in relativity theory it works and general at the video works in
quantum mechanics
is clearly something much deeper
about the conservation of energy
than just what we're about to demonstrate
but stovall swiss tarps rope
and i gave you an example
couple of examples where energy conservation fails
for example if this friction forces of course uh... energy conservation really
does work
are it works if you were to follow the heat
and everything else that's produced by the sliding your racer
but did not all i can just think of friction forces slowing the erase it
down
it would seem that the energy connecticut maji the energy of motion
suddenly goes away
and it's 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
car give another example other examples the spiral staircase
delivery of this way
supposing you have a force law
are that always pushes you are around a circle in the same direction
before it's always points around the circle
let's not worry for the moment whether centrifugal force takes you off the
circle let's separate pan
that there's a good solid barry a
another force which keeps the particle for 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 particles going to get faster and faster and faster
and it's clear who will by the time it comes back to the same position
there will be going faster
there's no possibility
of a
army
of a conservation potential in connecticut a g
you'll come back to accept exactly the same spot
so therefore the potential energy if there is any will be the same but the
connecticut igi has increased
it'll keep increasing
a forceful like this
is again on physical it never occurs in nature
the forces that occurring nature or core conservative
and they are always to the following form
the components of a foursome speaking about a single-particle now will
generalizes too many particles in fact we're generalizing arbitrary systems
but let's take the case of a single particle
before us and now think of the force as a vector
and drawing it into their mentions it could be in any number of them engines
is equal to minus
the derivative
alright vehicle right formula that works for you it means
this formula
triangle is simply a symbol for the residents
partial derivatives
magic that the potential energy is a function of position
and if the position with two dimensional as it is here
it would be a function of x and y
this is the potential energy
off this formula really just means that each component
of the force
rick burke x component of a force
and the white component of the force
are equal time minus
the derivative
in this case with respect to x
of your that some warning
and minus the derivative with respect for wine
of your tax and why
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 a force in the given direction
are proportional to how fast
the potential energies changing along that direction
our lips
use this
to prove the conservation of energy for the motion of a particle
i did last time but i've
as i said i was getting a little bit tired sore good again
and just do it as review
let's define
the quantity the total energy as the some of connecticut image eclipse
potential energy
okay what about what is connecticut a g i'll define it for you now
by definition
for particle moving its one-half
a mass of the particle
harms the velocity squared
for velocity squared means
if there's not more than one component of a lost city
it means the sums of the squares of the components of a velocity so in this case
with good intentions it would mean one-half
m
the exco aired possibly one squared
more generally i could write it as one-half the mass of the object
onto some over the components
of the ice quarter
levy x quickly why square plzzz east quay road
or whatever the number of directions of spaces
left the kinetic energy so if that's usually called t
began chris like
p stands for momentum he stands for kinetic energy don't ask me why
and then
the other contribution to the energy
yep before you would
a lot of people use vpn potential
but that gets mixed up with voltage
which is potential energy divided by uh... charge
so i use your steer clear of the for other people use the
it also gets confused with velocity
feet plus u
that is the total energy
i want to prove that's conserved as a consequence of newton's equation
that's because i made
what does it mean that it's conserved
it means but it's time derivative is zero
so or we have to do
is take its time derivative
used newton's equations
and prove that everything cancelled out when we take the time that's the
exercise is not very hard to exploit
okay so let's calculate the prime derivative of the connecticut g
revenue or the respective time of connecticut g
that is here it is right here is the key is the connecticut a g_ one half of the
squared
and is not changing
masses fixed
one-half as mark changing
the thing which might changes the velocity
august of course is the sum of squares of velocities
sources some
and uh...
v sub army
times the some arent by deepti what i've done
as i've said that the derivative of a square
namely the derivative of the squared
is v times the derivative d
the relative of a square of something
is the something
tons of time derivative
wilkes
i threw out a factor of get it right but i for the bins they are right
derivative square is twice the something times of derivative something
that gets rid of the factor of two
so decline derivative of the connecticut a g
is the song
overhaul of our actions of space
mass
times the ice component of the velocity
times now what's the what's the v_ survive a debate
that's a component of the acceleration so we might as well call it the
acceleration
sabari
component of exhilaration
let's calculate
prying derivative now
of the potential energy potential energy is also and is a function of position
when i arrived here that's a function of x
it could be a function of x_y_z_ all of the court next
what is it derivative u_n_
with respect to time
hillbillies with time for the same reason that it did before
because the position varies
if there is a little bit if you move in the ecstatic to live there is a little
bit if you move in the white direction if there is a little bit if you move in
z direction
and so in fact the derivative of u with respect to t
is a sum of term
during each time being the derivative of the u
with respect to atx
part of the time derivative of x
plots
d_u_ by d y
time to time to review two y and so forth
or i can summarize it
by saying
it's the derivative of you
with respect to the alright coordinator
x
times
velocity
in the eye fare action component of the velocity and yard direction
again son bogart
so now let's write down the total time the rhythm of the energy
some of these two
derivative kinetic energy the river of potential energy
and we can write down
time to read energy
this equal
subnational of all of our actions
let's take the use of are you first
and then write and any safari
first term here
you can see where this is going and grouping and with a survived for a
reason
and then
we have to add to that
again some of the harley
and it is
class
d_u_
brady ex-army
well sometimes beyond
both these terms
the loss of each of them
so let's take the velocity outside the parentheses
marks from
well we have
what's an acer body
supports
mass times acceleration
so this is the component of the force in the art direction
what's deal by the xie
that's minus
the force in the right direction
greatest right here
so this is just f
minus at
efforts in the bracket here
notice
the force is minus the director of the potential energy
all primes p
but f minus f zero whether it's multiplied by the or not
so thrilled by term
or you're getting zero
upshot
energy is conserved
and the only talking about a single-particle moving in a force field
but we could be talking about any number of particles
instead of of thinking of the summer
as just a somewhat with the three directions a space
they could also include the sum of a 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 conserve energy
no matter what the forces
as long as the rulers that the force on every particle is minus the derivative
of the you with respect to which position
so that was a review
remember if we talked about uh... conservation momentum last time
not store
i will now begin reviewing what i did not the last time
but which i had intended to do
towards you conservation momentum
buzz i said this was a little bit of trickery in a little bit of magic and
you might say
we are but why isn't you really concern
we will come to that
but for the moment would just observing
that newton's equations allow you to the following a certain things
namely
he plus v
the called energy
okay now let's come to momentum
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
before us
again
uh... we can do it in components absa body
is just the quartermaster
hirings
derivative of v some army with respect to t
again are a simply stands for a one of the three coordinates of space
mhm record
menace
doesn't change with time
so i can bring it into the kinda reparative
and just right before us
i'm not going to bother putting the indices down here
is derivative with respect to time
of mass times the
and now give mass times via name
the name is momentum
mast time velocity as a vector it has components
as well we correct this for each component
are in times these called momentum
until another form of newton's equations
is that force
is equal to the time derivative
of the momentum
i would object
momentum is called p
and so we have
we can make this a vector equation if we like
for specifically the time derivative of a moment
i don't know idon't think newt next week the following 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
or what assumptions only making about time well
no we didn't talk about it where it hurts
harm receiving that there's a universal concept of time
at every point in this room we could have a lot truck clock each one of your
clock to synchronize
and we're making the dumbness simple words the tony and assumption
that time isn't absolutely universal thing
at every point of space weakened synchronize our watches
so that everybody will agree upon the time
and that time just flows uniformly
uh... and uh... 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 pine
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 they were just a single particle in there was no four
sonic
momentum would be conserved because who would say the time to review the
momentum is equal to zero
so for a single particle would not forces on the balcony outer space
far away in math
between the galaxies
the momentum is conserved
at least of the approximation that we had more forces on the particle
but that's that's hardly uh...
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 a look and 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
before sun every object
is this some of forces
dual-core all the other objects
i saw the clothes particle one
than before some particle one
is equal
to look for some particle one
booth the particle to
coretta foreign film
plus
perforce undp article one
due to particle three
we assume that a particle doesn't exert before some itself
plus
eft form for
and so forth
for some particle one is the son
one
same thing for political too
before some particle tomb
is the force onto
due to one
because the notice what i've noticed my notation
when the patient is the first entry
is the particle that we're talking about whose we're talking about the force on
the first part of all the
the second entry
is
refers to the particle of the forces do tool
so it was a force on one
due to two
here's a force until blue one
plus the forces on too
do three
plus the force on to reform and so forth
let's just take three part that or right-to-work recused
from kate it pre particles
and finally
the third particle
or have forced due to
force on three who won
plus of force on three
now news assumption
law
contraction in reaction of course
is that the forces of a particle on another particle
forces between a pair of particles
or equal and opposite
so the force on water
do the two
might be an attraction for example
is equal and opposite to the force on field one
so if true is pulling one
upwards to the left
one is going to downward to the right
with the same magnitude 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
will come back to that
important point
is that before someone do it too
is minus the force onto a good one
we can make a victory could questions out of these if we like
they are prepared questions
and what the vehicle a what is known as lost any
it says that f_ one to
is minus f two one
f_-one three is minus f three one
an f three two
is minus f two three
okay now let's see if we can explore momentum conservation
well we'd like to prove is at the time derivative
of the total momentum of a system
zero as a consequence of this
what i mean by the total momentum i mean the total sum of momentum on all the
parts of the system all particles of make it up
what's right and i want to do the three particle case
d_p_ one
by dat
i'm not going to buy the putting vector signs above the room for the uh... think
vector
p p one by detain
is equal
f_-one
which means f_-one to plus f one three
possibly others uh... if there were more
or about the p to buy deepti
that's equal to ask to one
plus f two three
and finally d_p_ three by d t
is equal to f
three one
plus f three two
now let's and a lot
p p one by deeply deeply to buy deepti the petri by duty that gives us the
total time derivative
of the total momentum
so time derivative of the total momentum vp by detainee
this is just some of everything in the right hand side
but they cancel in pairs
at forty two was minus f two one
just saying the forces cancel
form three as mine s f three one
and have two threes minus of concurring
you know i represent no
and it says that the problem momentum pa pogo momentum of any closed system
into system would know what stone forces aren't
of course we might have are the forces
due to some other object in the system
then we will have to take it into account also
but we have a closed system a closed system means we've accounted for
everything in the system
the can exert forces
then the prime rate of change of momentum zero
and momentum is conserved
again
it comes from a rather specialist sumption
about looking for the tony enforces
but it's far more general
it applies to light
there's a light gray
bangs into the wall
it'll give the wall a little bit of portion change the momentum of the wall
why because the light grey has some momentum that applies in quantum
mechanics of the flies in the general theory of relativity
so it's far far more general in this little example shows
on
and our object
is to will eventually be
write the laws of physics
particular mechanics
in a form which is so general
that we can see why these things have to happen
okay any questions
in the tony in physics cities
all right and classical mechanics the pony in physics
and it only in physics means beget relativity
in the tony if they are good let's put it this way
according to tony in physics the speed of light is infinite
you turn on the lights in the room from lights up instantaneously
there's no elected
in the real world of course
forces do not packed instantaneously it takes time for force when u
when you do something to a particle overhear it takes time for through the
forced to respond over there
that's because of relativity special relativity rezoning
but only in physics
and the tony in physics means the forces of instantaneous
so yes we're assuming they're instantaneous
again
are
what's killing its
is that special relativity
undermines that instantaneous nissa forces
but it does not undermine
the conservation of momentum
sort some how much deeper
then uh... that are examples indicate
here
skis you could drive
uh...
when
but there will not make them was in the research
how many momentum laws are there for a single particle for example or how many
momentum laws are there all together for a combined peregrine system
three one for each direction the space
harmony energy laws
so altogether there are four conservation laws
three for three for the three directions basin one for the direction of time
and in the special theory of relativity they form a complex of four
related a conservation laws
but uh... this this
it's more general
okay
now its
my usual run
we're going to be a little bit of mathematics
not heavy-duty mathematics fairly light
but uh... what suspend a little bit of time
will do next is a bit of a wand too
what really is the deep law classical physics school principal or law
of least action
and it contains or was drunk
the law of least action
has all of the stuck into it plus a lot more
but arms before we do
we need some at
elementary mathematics extremely simple
math that you know very well
azwar spell it out
and remind some of you who haven't done calculus
sephora for a couple of years
so that or on the same plane him
the first thing has to do with minimizing functions
this is very elementary are not telling you anything you don't know fidel
function are several variables let's start with a function of one variable
fraction of warranty
some alone for about four supposes that if there's not stand for force now to
stand for function
at the y
and why is just some generic independent variable
initiatives
is the functions plotted
vertically apply f
horizontally i thought the independent variable y
and it has some shape
the first fact is if you want to find the local minima local minima
of the function
mean the places where if you're there
if you're going to find a direction
the function increases
a global minimum
would mean the absolute minimum almost possible minimum
but when i speak about local minima local minima mean
if you are confined for that region
you would think that the function was minimum right at that point
what does the mathematical condition for switching from a number
derivative or function should be equal to zero
that's not
sufficient
are
so first condition and he's used one
is that at a minimum
the slope of the function of zero
that says that the derivative of the function is equal zero minimum
it's also zero at maximum elders
the first condition is that the f_b_i_ d_x_ i_v_f_ but you are you
focal derivative in this case
poeple derivative it's only a function one variable
musical zero
now there are other conditions
they're out the other conditions are basically they have to sort out
two other cases
the maximum
the f five e y is also zero at maxima
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 gender generic term
for a function
which is either
maximum or minimum and you don't mean that may not know which
extremist one word
stationary in fact in fact is even another possibility
an inflection point where the function is flat animals down like that
again zero the river at that point
excuse
is a function which has a uh... uh...
an inflection point like that at the origin
it's derivatives also zero
at the origin
knowing that they wouldn't do it in a busy wrote dozens sort out these three
cases of minimum
maximum or inflection point
the word free the minimum or maximum
is extremely
extreme point
what's the word
anyone of these three
stationery stationary
station functions stationery if reports
which means
that if you move a little after more with the right
at night it increases nor decreases
well-being creases are at the second order but it doesn't increase the
principle of the derivative zero there it said to be station
the real principle of least action is really the principle of stationery
action
we'll call it the principle of least action and always pretend that we're
looking for a minimum of things
but in fact it's more general
principle of
is the principal stationary action
but i'm not going to draw that distinction mountain
after this point now
condition stationary point
is that the derivative of f with respect to wise equal to zero
all often say minimum
but i will mean station at that point
now supposing we have a function which is a function of several variables
effort x and y
and let's say i think about it let's draw the ex-wife plane
at every point on this plane function takes on some value
we could describe the function bias or contour map
washing comprar maps you know if they look like
are
beach contour
describing
a particular value of the functioned as you move from one point one come to a
point to another contra point the function changes
well arlene i don't need the contour map or we need to do is to remember that the
function has some value every point
on this to them regional playing
now supposing what i want
is the minimum
local minimum
with respect to two variables
what that means as a point
with a function is minimum no matter what direction you move warfare
that could be any number of the mentions
i'm interested in point where the function
achieves a minimum
which means that any direction the chemo warfare
the function
increases
however you move
because the bottom of a bowl
yell it with a strictly speaking with parking stationary
but i'll use the terminology a minute
and if you want to think about it uh... think about minimum if you want you
won't be making any other serious mistakes because in fact almost in every
almost every case in mechanics
are we really are talking about the minimum action
uh...
before to the conservative it's not a maximum
i think that
probably not through terrible loss independent forces like magnetic forces
although i've never checked that stroll
non perturbative
none conservative no no no no no
magnetic forces are conservative
but they're different than nature than uh... let's not get into but the moment
will come back to them
i believe there are cases in physics where mid actions not minimum but
they're unusual so let's just a bit of
minimum minimum what's not there
let's not worry about that subtlety
well first of all let's suppose
we are
they have a minimum right there and let's look at the function
what i meant to go right to 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
because of functional 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 acts
to be called as well
beleaguered say exactly the same thing about moving on the vertical axis
as a function is minimum at that point they will be minimum with respect to a
vertical
motion
so that serves the function
should have no derivative with respect to warranty
develop function of a whole bunch of variables
let's give them the generic name exabytes now
function of xa bari
the condition
that you had a local minimum
the second derivative of f with
with respect to everyone of the xa bodies
every single one should be equal to zero
let's do an example or to just go through
difficult for the folks in one variable at we don't need to do an example let's
take a function of two variables
and find its
local minimum
just in fact in will be global minimum
and just arbitrarily making up a story for us right now
uh... yes
xc where plus y_ squared
are up for three x
plus two y
let's put in plus x_y_
plus seven
that's my function
i want to find out where that functions minimal
so i differentiate with respect to expand set this derivative equal to zero
that gives me one equation
do the same thing for what it is me a second equation
one of those equations four
there for x and y they tell me where the minimum is it to you
so i have to equations for two or not
let's welcome our just as an example
are
the derivative the partial derivative of f with respect to x
the f_ buggy at
all right that's from this time we just get y
from this time we get plus two x
this term doesn't depend on x altogether
from this time we just get three
and there were no other terms which depend on it
so this is the derivative of f with respect to xd
pretending that why which is the constant
that's a partial derivative
and our role is to set the sequel to zero
that's one equation for x and y
let's do the same thing the f_-twenty wine
all right the f_-twenty wise x
points to why
plus two
equal zero
here are two linear equations for x and y you can sit down and solve them
notice that the seventeen command political
because the seven is just behind overall height of the function doesn't change
with a minimum is
so
for european sirs
excuse equal minus eight birds
and warriors equal minus one third if that's not correct
or
it's not important cancun
has again disappeared
lustre paper that i did and
that's not important
there is an answer to the problem is you can see it's just too plays into the
dorms
if you have in unknowns another words if you have a function of in variables
then there were any questions like this
and equations friend unknowns typically will have a solution
not always
some functions don't have minima
for example if i took a function
it just was like that
forever and ever it wouldn't have a minimum
they would just be no solution there would be no point in which the
derivative vanished
are sometimes more than one solution
for example but at the function with a few minutes
you'll have enough equations for enough funds to locate solutions
that's that's
mathematical point number one
you find stationary point for minima
by differentiating function or another way of saying it
is finding points where you feel very the function a little bit
it doesn't change
it doesn't change the first order in the variation basically
proj so that's that's a stationary principle of stationery functions
now will not going to be interested in minimizing functions of
a finite number of variables
we're going to be interested in something which is the face of it seems
far more complicated
it's minimizing expressions which depended not on some finite set of
variables
force themselves depend on the whole functions
hopes reject a res
whole trajectories
minimizing quantities which depend on a whole trajectory
before we get before i give you an example
would just say some words of power
the basic problem of mechanics is
lately
more than that the basic problem of classical physics but let's say
mechanics
the basic problem mechanics is to determine
the trajectory of the system
from its initial starting does initial conditions
that's a mathematical problem that's
of course in every case we may not really be interested in determining your
if we have ten to twenty third particles
determine the trajectory of that system is equivalent to determine the project
the reasonable confirmatory food particles
nobody tries to do that
instead we may see make some statistical misstatements about it or somewhere
some approximate statements
but visa
very precise question of classical mechanics
is determining the trajectory of the system
now what is it me
we spoke last time a little bit about generalized coordinates
generalized coordinates are simply
the coordinates of the system they could be the cartesian coordinates of the
positions
of a system of particles
they could be angles
they could be the or re inflation of this array sir
they could be any number of different clients are of
coordinates that we might use for example on the surface of the earth we
might use latitude longitude to locate them object
those are all generalized coordinates
the there and they may not be cartesian coordinates that may be angles or other
funny kinds of variables
but
they do serve
ripple locate an object in space
or locate a system
in space
both are called the generalized coordinates and the often labeled by q
your own one proved you and
for their first single-particle there are three kilos
x_y_ insane
fertile particles theorists excuse
the x_ y_ and z_ of one particle in the x_ y_ and z_ of the other particle
effect april twenty third particles
there are three times tend to the twenty third cues
but as we've seen already
the cues 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 becuase docs
the prime directives were
you need six
pieces of information
plane 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 q_-one of t
acute to obtain
to free of pain
it means to determine all of the coordinates as a function of pune
all of the poor particle coordinates or the angles or the only impatient 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 any system of functions as a function of time
we can call a generalized trajectory
generalized trajectory
week alone
imagine how resign fully
plotting
all of the cues
now i was up we have just drawn to the mentions here
two dimensions will be a stand in fact any number of
ben mentions q_-one through q n
and vertically we plot the time
but reject their e simply means exactly what it says trajectory through the
space
trajectory through the space
is of q at as a system of queues at every value of time
the mathematical problem
that that's put in front of us body
the questions of classical mechanics
to what extent
can we determine the trajectory from the starting point
alright answer of course it is given newton's equations we can
the kiltie
not just from the starting point we need to cues and accused opt to start work
if we don't accuse an acute docs
we can determine the trajectory
the equations of motion newton's equations of motion
what they tell us as uh... every point the longest reject three
they kind of tell us if we're moving in a certain way
how to move net
they tell us that if we know the velocity along here
are sorry if we know where we are along here and we know the forces
than we know the acceleration
so that tells us if we know what we are and we know where we're moving
we know will be next and we know where will be moving that
waco say there's
is that newton's equations local equations along the trajectory
and they tell us information they tell us how to proceed from one point who
knocked to the next along the trajectory
you'll 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 xd
so their local
equations
and that's why the differential equations differential equations are
equations among positions in the derivatives and so forth
and that's the kind of information that they have it
now of course if you know this information all along the trajectory
then you can build up the whole trajectory
but the equation itself isn't equation which tells you local information
the principle of
least action is another formulation hussein physics
which looks at the trajectory of the whole
instead of being given
the position and the velocity
single-cell time
which have given is different kind of information
you've given the position at the beginning of a trajectory
okay it could be the position of this array sir at time t equals zero
after a certain specified amount of time
you're given the position of the system at the end of the trajectory
another words you're giving the two and point of the trajectory
so for example
here's an example here's the eraser
i tell you at time equal zero it's over here
i'm not going to tell you how it's moving
so i can tell you what's good the next but what i will tell you is that after
one second
it will be over here
resolving one project three
which goes from here
bahir in one second
there it is
now by phone or will also be
we 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 and a single point along with your
jected saying the mouth of information
instead of giving the position and velocity at one point
i give you the positions
tools separate pines
that's the problem
at another formulation of the of the
question of mechanics
instead of the initial data problem
the problem of finding a unique trajectory which connect two points
in this forum
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 of a whole trajectory
the some quantity about that reject ary
minimum
for the true trajectory half of the true trajectory
and will be larger
for every deviation of the trajectory from the true trajectory
true drip
correct directory
there's an example
had two points
for simplicity we could take them into their mentions
calculate the distance between them
and asked what is full line we're not too distant junked draw a line drucker
between them
rancor between
calculate the length of that curve
and then ask what curve
minimizes the lane
what curve
gives you the smallest
distance along the curb
between those two points
of course we all know the answer to straight line
but what's the mathematics that goes into it
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
swords any much more complicated object
calculating the minimum
or functional a few variables
this two approaches to it
one is the local approach which will show you what it means in a moment
and one of the global approach
yeah let's talk about their local in the global approach
global approach is just what i said you pick your two points
you calculate the distance along the curb
at some quantity that depends on the entire curve
and then you ask what minimizes going along that curve
that something so very global view
where you'll minimize something associated with the entire career
local point of view
goes as follows
supposing we know let's focus
let's go take a little microscope era and focus along some was along with the
check the rear over here
and fact raw purposes let's imagine breaking up the trajectory into a lot of
little pieces
each of some small way
let's focus right in here
the curve comes in over here sort pin 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
to all of those little pieces
and ask where should you locate
the midpoint
in order
that the curve
the as short as possible
now the answer doesn't the pens
where you put this point
does not depend on the rest of the curve
if your happen to know this point the map point
you know
the point over here
and the point over here
the rest of the curb doesn't matter
you will have to in any case u
if you want to make the crevice short as possible
and you know that passes through here in here
then all you have to do is ask how do you make that little segment between
short as possible
what's the answer
into 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 is a short as possible
go straight ahead
we could express this is a differential equation we will uh... as different
equation
which basically says that the
difference in the slope from one point the next is you're wrong
but it's quite obvious that what it says within this little red region here
is just go straight ahead
if you generalize that will occur bill will tell you at every point just go
straight ahead
so this is the local version
which
says gopi each point and 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's a straight line
are
so there are these two different versions
of beginning
about the principal of least distance this would be the principle of least
distance between these two point
let's see just uh... for future reference
worked out
let's uh...
let's try to define our problem mathematically even if we won't solve it
yet
stone
fine i saw before you were told you what the answer a straight line
because you know that anyway
but let's do a little bit of mathematics and state the problem in america medical
form
okay let's say a
let's take up current i think i'd like to get free japan
let's think about crew
as existing in the ex-wife plane
let's suppose we know that it goes to this point
at some point
x one
and another point that extra room
and let's think of wayy as a function of that
some curve
and what we know
is the value of wayy annex one
we know the value of white x too
molars i tell you with the in points are
and i ask you
the find the shortest curve
well the first thing we're going to have to do
is calculated the distance along the curve for any curve
and then minimize it
are so let's imagine
let's not imagine let's calculate distance along this curve
will call this function why elects
so if you like
this is a problem of minimizing something with respect to a whole
function
finding the function
which minimizes something
let's calculate the distance along this curve if we know why effects
first begin with a little segment
a little segment
uh...
size d_x_ only expect
right so it's a little
gap in the function it's a little
segment in the function
over
the interval the acts
let's uh... let's drawer over here
your ears
goes over distance the acts
wire xx
and we can also speak on the wind
use b_-one
as we go from point xd exposed iacs
the function goes from wayy to wipe plus d_ y_
what does the length of the high partners of the triangle
if we end up always little triangles aneta paula jai pot what's the plural
papa news
i don't know
i guess so
if we add up all the
hard copy
mattresses
if we add up a lot of partners is
will then
compute the entire curve so let's compare
well this sign is the wuyda size the x
the destined for a long he'll park the s where this is square of the distance
and stands for distance again one of these letters that has nothing to do
with over there were
s stands the distance
the square of the distance along this had partners here
is clearly d_x_ clip lusty why squared
or protect the square root it tells me
distance
along our partners
let me write this in the following way
reich dscn take over here on the right over here
is infor
that the acts factoring out the acts
times square root of one
plus
gyi by the x_ square
when i get that far
if i bring the d_x_ inside the square root i have two square it to bring in
inside the square root
then the one will become the exco erin
and this time he'll just become b_y_ squared
so i just like that our of the acts i want to think of x's the independent
variable
wide by d_x_
that's tested the repetitive of the function y impact
so that's nothing but the function of the return of the function if you'll
provide gave you a function wayy
you could differentiate it
and this would be it wide by the axe
all right now let's is up all of the little intervals
and not believe the roles means doing an integral
on the curve
from point x one the point x two
escort s
between one and two
is equal to the integral
from x want two x two
of justice quantity here
square root
of one
delighted by the x
square
the acts
i'm not going to do anything with this right now
just wanted to work out for you in terms of why iraq
if you know if somebody gives you the function why have x
this is the way you calculate the total length along here
without problem
our problem is the find that
function why effects
which makes this integral as small as possible
now remember when i say that it's a small as possible
are
works inside the square root is bigger than one one-plus something positive
you might say okay look
if i want to make this a small as possible i will just uh... make uh...
what's inside the square root of u_s_ more is possible
and make the wide the exit poll to zero
now that's not the answer to this problem because we're making an
assumption
the assumption is that the current goes between these two points
fifty one by the x_ resume our everywhere as
we're just before that
wouldn't go between these two points towards
given at the curb goes between these two points
we want to find the minimum value
of this integral
the minimum value of this integral and that means that curve why have x
which does that for us
order function wherever acts
in this case if we thought of x_ is kind of like conning
and why is the coordinator of a system
this might be determining the trajectory
by some funny principle
of least distant in space and time but for the moment
just wallace's is
non-performing the shortest distance between two points
no i haven't told you how to do it
will get to actually how you do it but that's the mathematical problem
find the function which minimizes
this expression
uh...
this s here
depends on why you know that
you could sort of think of it as a function of a function
given why impacts
he is a rule for calculating a number from it
it's a function of a function it's not a function of a few variables
that's a function of an entire other forms of sorts of number
which depends on the
empire function
this in the
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 as a functional of
wire backs
problem is to find the minimum value of that functional
subject that's the some constraints that passes through
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
asia
principle of optics
is the
principle of lease pine
light grey goes from one point to another point
along the path
that minimizes the amount of time that it takes for the uh... for the light
rate the go from that point to another
remarks principal how we figure that out i don't know
uh...
is the correct that
now
you say that just means
like those in straight line
it minimizes the time that it goes from one place to another like those of the
speed of light
that must mean 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 more interesting kind of set up
we might have a material of some sort
with an index of refraction which means the speed of light which changes from
point a point in the material
then that's not so clear what subject three to take in order to minimize the
time there take so let me give you an example
let's suppose
we have a material three-dimensional material areas
and
the speed of light varies for simplicity
there's nothing that there's nothing special about this but the simplicity
let's suppose the speed of light just varies with the vertical axis here
so we can provide this up into and
uh...
ribbons
and these ribbon the speed of light is different
for example the speed of light might vary from some minimum
the sun large value up at the top
like a slow down here
and they simply it goes to save a lot of the same velocity no matter what
direction it's going in to assumption
there are materials
well-lit moves with different velocity in different directions but i'm not
doing that
art so far
if the light is going this way it moves with a certain la city of going this way
most of the same velocity
the velocity does vary from strip the strip
now i want to know
what's the path
sold at the time between this point in that point
is minimum
that's the true trajectory of the light rain
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
calmed the path opera little bit i didn't mean to curry quite that much but
let's carving up that much anyway
then
it's going to spend more of its path moving with the speed is faster
so it will want to get the region's want to get
like doesn't want to do anything
but in order to minimize the time that it takes to go from here here
it might want to get to any region quickly
with a velocity is larger
and then make most of the path with a larger velocity
but now obvious that goes in a straight line and it doesn't go on a straight
line
effectively going to get three something like that
if our 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 upload part of the jury doesn't get you
anywhere
there was no good to just go straight up to go straight across
the some compromise
we you curve the trajectory to get into regions quickly with the velocity of
light is faster
but not straight-up and straight across
axo what is then the
the quantity which are trying to minimize
when you speak about the principle of least time
will work out for you just for fun work it out
again we have some solution
which is why iraq so we want to find live acts
the path
which minimizes the time between two nonpoint
notices a theme here
were always
talking about tool known points
and asking for the trajectory
or the function
which interpolate 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 or another
first of all again will break up the trajectory of the swollen lymph ables
each interval
has distance which we've already calculated
here's the distant
tillis write that down again over here
this wednesday s
is equal to be a
times the square root of one-plus
the wide by the x
derivative square
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
rollers that the espy deep feed
is equal to the varsity is equal to the speed
let's call that's
but she is not a constant
it depends on walking
i could also take at the depend on experts such as for simplicity
i'm imagining this material
which is a kind of a lyric material
with different speeds at different parts
see as a function of wayy
well that allows me now to determine how long
politely racist then this
in each little interval b x
we just saw for the team
so detainee
is equal
two bs
provided by sea of wine
all i've done
is mm is solved with the t
multiplied by deepti and abide by sea of water
so that's how much time
the like great takes to go from one point to a neighbouring point
where they're separated by distance the s but no
here's the s
so we can also write to us
as d_x_
square root one plus the wide by the year
squared
all divided icy of why
that's the taryn in terms of the function while you're at cs
retort
we have some function why have acts and we want to see how long it takes like go
along that function
the why buy the x_ is just a derivative
if i gave you a function you could differentiate it
the xd
that's just a little acceptable from here to here
and see hawaii
is the velocity of light
along the curve at crate y
what's the total time that it takes to get from here to here
we and up all these little line and all these little intervals of time we had
them up which obviously means we do it in to grope
republican
between one and two let's call if you want to
is equal to the integral
the acts
from the starting point
x one
three and point that's true
start the end
square root one
plus the wide by the experts were
provided by sea of white
if i told you would see a while i was
and i gave you the function y id x
we could determine how long it takes
belie tree to go from here to here
now that doesn't solve a problem not just sets up our problem
it tells us for every curve why have acts
how long it takes to go from point one two two
then our problem is the flying the curve
which minimizes the time
obviously hard
rights or find the car such talk all possible curves that you could
so true
and find out what curve minimizes the time
both plugin is 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 u
have a
chain or cable or spring or whatever or waited strings
waited spring or we the people
home between two points
now we're assuming
that we know the length of the chain
length of the chains fixed
we might ask what configuration does the chain tape
and the answer always is good minimizes the as a potential energy
uh... we would
were called just tell you that as a fact that
it minimizes the potential energy
that means effectively get as low as it possibly can
because the potential energy is just a function of height
potential energy increases as you grow up
the tries to get as low as it can
well i can't drop down to the floor because the chain has a fixed length
are
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 them debate here
so the some curve
some function
i'll fix length
which minimizes the potential energy
laboratory with the function as a man remember what is it
netscape had refused to work at mary's acya
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
for the curve itself
all it's just the height of each other
need basically the height of each molecule
it doesn't matter because seller
because
you can take it from the fixed-point
but if you want to change zero of potential energy which has been a
constant
and it wouldn't change the uh... the answer
but uh...
but right now i i don't want to set up you know i don't want to set up this
problem all i want 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
infinite number of variables
the values of wayy at every point at
the mathematics is called the calculus of variations
it's a problem minimizing a functional with respect to a function
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
another words of a system where the bedroom right
nod one function
for a whole collection of functions
q_-one of t
q to of t
molecules are vivid generalized coordinates of the system
q_-three of ten e
the whole collection of functions which define a trajectory
the system
finds the trajectory
which means find the collection of functions
q_-one through killing of t
which minimize something
what is it minimizing what is the quantity
the quantity is called the action
that's the principle of lease faction that their exist a quantity called in
action
which minimizing it with respect to the trajectory is equivalent to the usual
rules of mechanics
in particular
equivalent the newton's laws of mechanics
but as in many many situations far more efficient than farm for a bit cool
are
then just writing down
newton's equations
their equivalent formulations of the same thing in which one is the easy and
more useful depends on the system
but believe me there are systems where the principle of these factions is a lot
more efficient
van nuys in particular
if i were to take this array so which is composed of ten to thirty of particles
and i have to write down the hall tenth of the thirtieth force laws between all
three times ten to the thirtieth coordinates
it would take me on long time to write down those equations
if i tell you on the other hand
that the trajectory that this object takes to moving through space in
rotating
minimizes some action
and i know how to write down the action
is a much much more efficient
pool for many purposes
rats
that uh...
armored
it implies that these laws in all cases where newton's laws
would be uh... the underlying data description a one on the line will in
the case when newton's laws apply
it there it
is equivalent to movies was
but it's much more general
that's more general for example an appliance the problems in special
relativity promotional objects a special relativity
and applies too
fields
it applies to the electromagnetic field
it applies to basically every classical problem in physics
not just motion of particles promotional waves sir view
the whole works
the principle of least action
uh... is the most general formulation of mechanics or
clapham classical physics that we have
so all systems and classical physics or
so i can think of
have a formulation in terms of the principal least action
if it's not particles but something more complicated
the principle of least action might not you don't mind we have to generalize it
but there are a little straps and a little steps total step we're taking now
speak about trajectories for a system of particles
but their own yes yes and of course
that is an example of using more general principles of
borrowers
work there are mechanical systems where you wouldn't get too far with that
neither and where the more general principle least action would be the
most uh...
efficient way to go about it
all right so
i will tell you right now with the action is
this section of the seem bizarre
it's going to seem strange
could be a little bit surprising
on
but the final answered have or whether it's the right action
is to test allowed for all new tony in problems
and see that the principle of least action gives the same rules as newton's
equations
once we do that will then have a reason to believe that this is the right actual
islanders
facilitate the growth of the trajectory so it's
it's now the independent variables time not x
independent variables time we have a whole bunch of queues of t
or axes of feeling of wood right
we calculate
the kinetic energy and the potential energy
and we integrated over time
integral
of the kinetic energy
which is what one-half
for ordinary mechanical system composed of ordinary particles
uh... ha
let's uh... just i think of it for one particle length simplistic one particle
in one dimension
and then i'll tell you of the generalizations one particle moving up
and down and access
so the problem in is to find x of t
find the curve or find the function except the which minimizes a certain
thing which i will now for the world is it's called the action
you might expect
if i tell you that has to do with the energies
you might expect that has to do with the connecticut igi one-half and be squared
which is one-half the xp p squid
plus the potential energy
integrated
overtime
another words at every point along the trajectory
you might write down
the and the
action
for a small interval of time
would be the total energy times bt now this is wrong
bizarre ory
the answer is
almost this but not quite
minus
the kinetic energy minus
the potential energy
that's called the action
uh...
the final in the final analysis
what do you call list plus something on my know something is of course a matter
of definition
you can you can read the fine in new variables you can call a w which is mine
is you
and put in
plus w here if you like
but then
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 no equations but you'll find it is a questions instead of
finding the force being
minus doubled b_w_b_ extra point plus the w_b_ exit you'll find out that the
thing that you called w
is behaving
like the negative of the potential energy
so you can call it anything you want
i will call of mine issue of x_ here
but this is the thing which is called the action
sometimes action is called ari
sometimes it's called s
i want to use s because we've already use vestal path length
and sometimes it's called a fraction
this is the one case
where i will use variable
which is linguistically
for medically or whatever the word is
uh...
sensible
what i suggest is we go through a bunch of examples
on
it drew
i'm throwing something i do now
and it is not you seek to explain where this comes from the warranty this is the
answer
the best thing to do what they
instead
examine it
and find out
then is the same as moons equations
and then we can go back and say why didn't have to be used to get there was
a questions
i rather not try to motivated point and just examine it and see what we get
so let's uh... what's
hold in suspension are questions about why this is the right thing
but let me know to some things about it
first of all
let's compare it
with the principle of lease distance of the principle of these time
principle of least time
also had integrals
now in those cases are used
x's the independent variable
and why is the dependent variable
now i'm using hinmen as the independent variable
and expands the deep end unbearable
so i pull the switch on your location
what i the independent variable which is like x
in the previous example
is now t
instead of trying to calculate wayy of x and trying to calculate x of t
so the deep end of variables x the independent variable is p
but let's just compare it with the principle of leases dot
of lease distance for example
or the principal at least time
principle at least distance told us to compute one-plus p wide by the axe
square
integrated the acts
if we wanted the principle of least time
the optical principle we will also throw in here on one oversee of
wacom x-axis
why start
first of all
the thing which i am computing here depends on two things
it depends on the why of x and it depends on the wide by pierre xx
so it depends on the function
and the derivative of the function with respect
independent variable
that's also true here
this depends
on the function
exit fee
and also
on the first derivative
of the function with respect to the independent variable
so what does the same kind of penned it doesn't have a square root of it
but still
the faith in the into grand here
depends both on the first to review
of the function
and the function itself
depends on what effects and it depends on the wide by the acts
likewise here
it depends on x of tea
throw the function you
and on the exploding
so it's a similar kind of thing
it's an integral along the
independent variable
from one time to another
we will also
put in an assumption which is very similar to what we put in the principle
of least time
mainly in the principle of waste time
we said
the integral is between to fixed-point
which we don't vary
we hold the two points fixed and we ask what's the trajectory which takes the
least time between is
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 peace zero
and x at the end of the trajectory
not to be changed
so it's a trajectory between to points in space and time
lightbulb gratuity and analogy is
as i said
the basic idea is instead of giving
the position and velocity
you giving the position two distinct time
prying the trajectory which minimizes
this integral of the connecticut a gene minus the potential energy
but very surprising it even still surprise to me after forty years of
having studied this for more than forty years
and i know the reason why it is i'm still always surprised
that it's the connecticut jeep minus the potential g
you should also be surprised at the connecticut minus the potential but it
is
that's the action
the thing in the
integral here the integrated
is called the legrand here
the connecticut d minus the potential energy called boulevard
general form the script alle
it's a function
arv all of the cues
that's a function of a killer fees
mccue doctor kings
the action is always an integral all the time
from the beginning of the trajectory to the end of the trajectory
the integrated
depends on the coordinates
x
and the velocities
xbox
occu thank you dot
that's the jew most general version of
the principle of least action
namely that their exist something is the law of physics
law physics says that for every mechanical system
there exist a quantity called a lagrangian
i don't wish to build the action
and then you'll find the path
for a project they re
which minimizes that action
in simple cases in fact in all cases the correspond napoleon physics of one kind
or another
this lagrangian is always
the difference between the connecticut maji in the potential energy
and more general systems
which don't have a new tony an analog
this lagrangian may well be something totally different than peculiar
but will still depend on cue and cute dot
and
and the principle of least action will be more fundamental and anne card of the
cold war is a physics
in particular
the uh... this is in the totally rule
the motion of a particle in the special theory of relativity is quite different
than this
but it still has this form
well we're going to have to do next time we won't go tonight
is learn
to find given 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 the tony in equations of motion how do i
go back and forth
from a grungy in
the newton's equations of motion
and it is not newton's equations of motion what are the equations of motion
the follow from a given lagrangian
notice that this if we know how to do this
if we know how to solve such problems
we would solve among other problems
by the same roles
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 principal
elise time and optics
methods
the techniques are exactly the same
so the next time will work out
what are the equations of motion
which follow from a particular action principle
in program is copyrighted
university
inferred dot the