Kind: captions
Language: en
this is a video about a single simple
rule that underpins all of physics every
principle from classical mechanics to
electromagnetism from quantum theory to
general relativity right down to the
ultimate constituents of matter the
fundamental
particles all of it can be replaced by
this single rule feels like we're
approaching spooky territory we're
approaching spooky territory I hear you
and I agree
it may in fact explain the behavior of
life itself I think I'm stuck in some
classical mindset where to me the local
picture the differential equation way of
thinking about the universe that that's
really what's going
on but I fear that I have it exactly
backwards and it all starts with a
simple
problem if you want to slide a mass from
point A to point B what shape of ramp
will get it there the
fastest this is known as the problem of
fastest descent you know Common Sense
you might say take the shortest path
straight line from A to
B but if you bend the ramp down a bit at
the beginning well the mass accelerates
to a higher speed earlier so even though
it travels slightly farther it travels
faster and beats the straight ramp so
the question is what shape provides the
perfect balance of acceleration and path
length to minimize travel time according
to Galileo it was the ark of a circle he
showed that this is faster than any
polygon but is it the
fastest nearly 60 years later in June of
1696 Johan bruli set this problem as a
challenge to the best mathematicians in
the world mainly because he's a big
showof and he wants to show that he's
better than all of them he gave everyone
six months to come up with Solutions but
none were
submitted so gotfried libnet a friend of
beri persuaded him to extend the
deadline to give foreigners a
chance and I think Newton was probably
the intended target because everybody
thought of him as the best and so Yan
would have wanted to show that he was
better than Newton he was um no longer
really active mathematician or physicist
he was working as the warden of the mint
like a big high government position
and on the 29th of January 1697 Newton
returned home after a long day at work
to find Boli's challenge in the mail
irritated he wrote I do not love to be
dunned and teased by foreigners about
mathematical
things but the problem was too enticing
so Newton spent the rest of the day and
night on it and by 4 a.m. he had come up
with a solution something that took
bruli 2 weeks
Newton submitted his solution to the
journal philosophical
transactions he sent his solution there
but didn't sign it and Johan beri after
seeing the solution is alleged to have
said I recognize the lion by his claw
you know that okay you don't need to
sign it Newton I can tell who you are
who else could give such a
solution and while overall Newton
dominated Bui in this instance bui's
solution actually out Shawn
Newton's I could see why Johan beri
wanted to challenge everybody because he
came up with a really clever I would say
like a truly creative beautiful
solution to do it he took inspiration
from a problem faced by ancient
philosophers how does light travel from
one place to
another this was contemplated by hero of
Alexandria in the first century
ad he realized that in a single medium
like air light always follows the
shortest path a consequence is that when
light reflects say off a lake the angle
of incidence is always equal to the
angle of reflection any other path
between the start and end points would
be longer but when light goes from one
medium into another like from air into
water it bends in a peculiar way it
refracts and it doesn't follow the
shortest path if you've ever dropped
something at the bottom of a swimming
pool and you look for it through the
water if you put your hand in there it's
not necessarily where you think it is
because the light has bent away from the
flat
surface so what is the guiding principle
here well over the next, 1600 years
people slowly figured out that the sign
of the angle of incidence divided by the
sign of the angle of refraction is equal
to a constant n which depends on the
nature of the to Media this came to be
known as Snell's law but no one knew why
it worked that is until
1657 so another great mathematician
enters our story at this point this is
Pierre FMA by day he was a judge but at
night he'd come home and hang out with
his wife and kids and then do what he
really loved the most which was work on
math for fun he mostly worked in pure
math but at one point he got interested
in the question of why does light obey
this principle of
refraction and he thought maybe hero of
Alexandria was on the right track but
it's not distance that is being
minimized but rather time but to see if
this was true for refraction would be
difficult he would have to work out
every possible path like could take by
varying the point where it intersects
the boundary and compute the time for
each and then show that light takes the
path for which the total travel time is
the shortest
he doesn't know how to solve it and he
worries that it's it's going to come out
complicated even if he could solve it so
he's not going to do
it I think it's because he wasn't super
interested in physics actually but
anyway years go by he gets interested in
it five years later and tries to solve
it and then he does solve it and he
shows that Snell's law actually pops out
as the minimizing path for light you
know under these conditions of moving
from one medium with one speed of light
to another with another and that
constant n well that's just equal to the
speed of light in the first medium
divided by the speed of light in the
second medium which allows us to rewrite
Snell's law like
this and he says this thing I just want
to read you a little quote because I
love it he says that this is the most
extraordinary the most unforeseen and
the happiest calculation of his
life see it's good to do
physics if you use that principle of
least time you can explain everything
that was known about light at the time
of FMA it's the first time as far as I
know that anyone shows that nature obeys
an optimization principle that nature
does the best possible thing in this
case that light takes the shortest
possible
time now bruli knew about fma's
principle of least time and he thought
he could use it to solve the problem of
fastest descent he converted the problem
from a mechanics problem about a
particle sliding down the choot to a
problem about
Optics instead of a mass that's
accelerated by gravity he imagined a ray
of light that would go faster and faster
as it went into layers of less and less
dense media and if you make the layers
thinner and thinner where Snell's law is
obeyed at each interface you eventually
get a continuous
curve now the question is how should the
speed of light change from one layer to
the next so that it accurate models a
falling object you could try to solve
the problem by thinking if the particle
has to fall from A to B it's going to be
picking up kinetic energy it's going
faster and faster as it slides down the
Chute and it's converting the loss in
potential energy into this kinetic
energy if you write down the
conservation of energy for that
relationship you find that the velocity
that the particle achieves at any time
having fallen a distance let's say y its
velocity squared will be proportional to
Y the height from the top so velocity
goes like the square root of Y and that
be sort of like saying imagine light
moving in a way where instead of a
constant speed of light the speed of
light is proportional to the distance
from the
top now let's zoom in and look at a
single interface we can plug in our
expression for the speed of light in
each layer into Snell's law then you
find that the sign of the first angle
divid by the square root of Y for the
first layer is equal to the sign of the
second angle divided by the square root
of Y for the second layer and now here's
the key Insight Snell's law also holds
for the next layer and there the ingoing
angle is simply Theta 2 so this is also
equal to the sign of the third angle
divid by theare < TK of Y3 and the same
goes for the next layer and the next and
so
on in other words this ratio must be
equal to some constant call it K and
this equation the Story Goes beri
immediately recognized as the equation
of a
cycloid that is the path traced out by a
point attached to the rim of a rolling
wheel this is also known as a
brachistochrone curve from the Greek for
shortest time and so the astonishing
solution is that the fastest way to get
from A to B is to follow an arc of a
cycloid not a circle a shape called a
pyoid now this curve also has another
surprising property no matter where I
release the Mass from it always reaches
the end at the same time for this reason
it's also known as the toone curve from
the Greek for same
time upon finding this solution bruli
wrote in this way I have solved at one
stroke two important problems an optical
and a mechanical one and have achieved
more than I demanded from others I have
shown that the two problems taken from
entirely separate fields of of
mathematics have the same
character little did bruli know he was
on to something much
bigger around 40 years later one of his
students Pierre Louie deeri also studied
the behavior of light and particles and
he noticed that there are cases where
the two behave very
similarly this made him think what if
fma's principle of least time wasn't the
most fundamental I mean why should
nature care about minimizing time maybe
there is a more foundational quantity
being minimized one that doesn't only
govern light but also
particles so in the 1740s he proposed a
new quantity which he called the action
it is mass time velocity time distance
his thinking went something like this
the farther something travels the
greater the action the faster it goes
the greater the action and if it's a
particle then the more massive it is the
greater the action if there are multiple
segments to the journey then the total
action is just the sum of the mass time
velocity time distance for each segment
to see the principle in action here is a
super simple example with no friction or
losses imagine A5 kg ball is rolled over
the ground for 6 M at 3 m/s then that
would be 9 units of action if the ball
then bounces and travels another 6 M at
3 m/ second then the action for the
whole trip is 9 + 9 or 18 units of
Action Now what maer claimed is that out
of all possible trajectories where the
ball bounces off the wall the path it
will follow is the one that minimizes
the
action in 1744 he wrote this action is
the true expense of nature which she
manages to make as small as
possible so what was the response to M's
revolutionary idea he was attacked and
ridiculed one of his longtime friends a
fellow physicist named Samuel kig wrote
that not only is your principal wrong
you also stole it from livets F who used
to be a close friend of moeri accused
him of plagiarism bad physics being
stupid and just about anything else he
could think of in fact he wrote a
32-page pamphlet just to mock moper of
course this may have been partly due to
rumors vol lover had an affair with
M but not everyone attacked him some
just ignored
him I've taken a lot of math and physics
in my life I think you're the first
person I ever heard pronounce it he
doesn't get much
mention all of this was terribly
stressful for Muer who was nearing the
end of his life and more than anything
he thought the principle of least action
would be what he was remembered for that
that would be his legacy but now he was
attacked mocked ridiculed and
ignored unfortunately this treatment was
at least somewhat Justified because maer
came up with his principle by kind of
just picking it out of thin air there
was no obvious reason why nature should
care about mass time velocity time
distance or even less why that quantity
should be minimized and mathematically
the principle of least action was wasn't
rigorous
either but there was one man who
vehemently defended it and that man was
Leonard
Oiler the first thing Oiler did was he
replaced the sum with an integral so you
could calculate the action while speed
or Direction Changed continuously and he
used this to find the path of a particle
around a Central Mass like the orbit of
a planet around a star solving this
meant that out of every possible path
between two points he would have to find
the one for which the action was the
smallest this is similar to the problem
Forma tried to solve only now instead of
changing one variable he would have to
vary every possible point along the path
which is infinitely many needless to say
this was an arduous task math had not
yet developed the tools required to
handle such
problems fortunately Oiler himself
invented a new method it was clunky and
timec consuming but it worked through
this process he realized that the
principle of least action only works if
the total energy is conserved and it is
the same for all paths considered these
were two conditions that maer hadn't
realized were necessary so Oiler
improved the mathematical rigor of the
principle he found two extra conditions
and he provided a specific example of it
working Oiler is an astonishing powerful
and not just great mathematician but
appears to be a good guy as far as we
could tell he he was very generous you
can still read Oiler and really
understand it he helps you he's
empathetic you know he was uh like you
are man he's trying to explain
stuff but Oiler was still far from a
general proof that would have to wait
for another legendary mathematician
Joseph Louie lrange
Joseph Louis lrange was a shy
19-year-old mostly self-taught but
despite his age he was already working
at the Forefront of mathematics
including with oil's new method in 1754
he shared his results with Oiler who
replied that lrange had extolled the
theory to the highest Summit of
perfection which caused him the greatest
joy but besides being worldclass
mathematicians the two had another thing
in common they were both huge proponents
of the prin principle of least action
and around 5 years later just a year
after maer's death lrange succeeded in
providing a general
proof is there any intuitive way to
think about action like I I feel like
there's an intuitive way to think about
force and there's kind of an intuitive
way to think about energy but is there
an intuitive way to think about action I
don't know I I want to watch your show
to learn I hope you'll come up with it
because I don't have a good feeling
about Action Now I want to explain lr's
proof but I don't want to do it the way
he did so instead we'll go through three
steps first we'll explain the general
approach Oiler and lrange came up with
then we'll rewrite the principle into
its modern form and finally we'll apply
this math to a simple example to show
you why it
works so first the general approach if
there are infinite possible paths how do
you find the one with the least action
well Oiler and lrange realized you can
do it in a similar way to how you find
the minimum of a function there you take
the derivative and set it equal to zero
and where the slope is horizontal that
must be the minimum so if you took a
tiny step to the left or the right the
value of the function basically doesn't
change and similarly if you have the
path of least action then if you were to
change it a little bit by say adding a
tiny bump here or flattening it out
there imagine we're just adding a tiny
function Ada to our path of least action
well then the action basically shouldn't
change because we're at this really
special point the path of least action
you add a little bit to the minimal path
but the action is still the same if that
is the optimal path that has the least
action then any other path must have
more action so the counter there is like
all of this is the first order so if
you're looking at linear
terms that are proportional to AA the
deviation then the first order the
difference in action will be zero the
way you could imagine this is like let's
say you're at the bottom of a bow and
you're at the minimum and you make some
tiny step away and we call that step AA
if that change would be proportional to
Eta you would maybe increase on this
side but decrease on this side and then
this would no longer be a minimum so
sort of the coefficient of ETA has to be
zero but since you're at a minimum it
kind of goes like a parabola so it can
be proportional to ATA squ or
potentially some higher order term so
there is a tiny deviation in the action
but it's not proportional to ET so to
first order the change in action between
the optimal path and some trial path is
zero so what you can write is that the
action of this trial path minus the
action of the true path is equal to zero
to first
order this is a compact way of writing
the principle of least action and it's
the general approach you use to solve
all these problems so with that in hand
let's rewrite the principle into its
modern form starting with maer's action
which is the sum of mass time velocity
time distance but Oiler changed this
into an integral so it's the integral of
mass time velocity integrated over
distance now the velocity is equal to DS
over DT which we can rearrange to get
dsal vdt and plugging this in we have an
integral of mv^2 over time but wait
that's just twice the kinetic energy and
as oer pointed out the total energy must
be conserved total energy is just
kinetic plus potential so we can rewrite
this as t = eus v and filling this in
for the second T gives us that the
variation of t + e minus V integrated
over time is equal to
zero now we can split this integral into
two and since the energy is constant we
can integrate this term over time to get
this and we can simplify this even
further just like with a normal
derivative we can write the variation of
e * t as e * the variation of t plus T *
the variation of e but remember as Oiler
found the energy of different paths has
to be the same so the variation between
them is zero and this term drops out if
we rearrange this like so then we find
that the variation of this integral is
equal to minus the energy * the
variation of
time this looks a lot like some other
minimization principle if only this was
equal to zero well we can make this Zero
by only considering paths that have the
same travel time if you do that then
there's no longer any variation in time
and this term drops out and what we find
is that maer's principle has changed
into another form where now the
variation of kinetic energy minus
potential energy integrated over time is
equal to zero T minus V kinetic energy
minus potential energy and then you
integrate that along a path that you're
traveling from A to B and then integrate
it with respect to time it's all very
strange and yet that turns out to be the
correct thing to
integrate now this is a little weird we
started with mass time velocity
integrated over distance and now we have
the kinetic energy minus the potential
energy integrated over time and somehow
both are ways to write the principle of
least action but that also means that
this integral here T minus V integrated
over time is another way to write the
action the first person to write the
principle of least action like this was
William Rowan Hamilton in 1834 and by
doing so he got the principle named
after him so the principle of least
action that we write as integral of L DT
L being the lran the T minus V the
kinetic minus potential energy they
don't call it lr's principle they call
it ham Hilton's principle so I guess
Hamilton is building on lrange in that
way Hamilton's principle is the modern
way of writing the principle of least
action and the way you'd find it in
almost every physics book in part that's
because Hamilton's principle tells you
how objects move from one place to
another rather than just giving you the
shape of the path two other important
differences between both principles are
that the action is now an integral over
time instead of space and a consequence
is that with Hamilton's principle you
now need a start and end point and also
a start and end time the third is that
with Muer twe's principle you need to
keep the energy of different paths the
same but the time can vary while with
Hamilton's principle the energies can
differ but the time has to be the same
between
paths so now we have our general
approach and the modern way of writing
the principle so let's apply it to a
simple example to see why it works let's
say I throw this ball straight up in the
air so it goes from some start point to
say a different end point in a certain
amount of time now if we call the height
of the ball y of T then we can plot
these two points like so and then we can
imagine infinitely many possible
trajectories that could connect these
two points some go a little higher some
lower some have Wiggles others don't the
only condition is that all paths must
have the same start and end point and
the same amount of time elapsed between
them now to find the real trajectory we
proceed as before we imagine that this
is the true path y of T the one with the
least action and then we imagine making
small variations to it by pushing it up
a little here down a little there and so
on making tiny changes at every time
step which we'll call Ada of T so when
you add Y and adaa together you get this
new trial path let's call it Q of T and
since the variations are small all the
difference in action between these two
paths is zero our next task is to solve
this equation so we compute the action
for each path for this we need the
kinetic and potential energy for each
and we write them as a function of Y and
Ada plugging that in you get the
difference in actions is equal to this
but wait this first integral is just the
action of the true path so these
integrals cancel and what you're left
with is just that M * dy/ DT * d/ DT
minus a * the derivative of the
potential integrated over time is equal
to zero we can rewrite this further by
using integration by parts that allows
us to replace this term with this
one and if we plug that in we now have
some function that when multiplied by
Ada and integrated over time has to be
equal to zero but since Ada can be just
about anything this can only be true if
this part is zero so what we found is
that the action is minimal for the path
that satisfies this curious differential
equation now it might look complicated
but it's not minus the derivative of the
potential is just the
force and the second derivative of
height well that's just
acceleration so if we rearrange this we
find that the path that satisfies the
principle of least action is the one
that obeys FAL
ma in other words the principle of least
action is equivalent to Newton's second
law but it covers more than just
mechanics forma's principle of least
time turned out to be nothing more than
a special case of the principle of least
action so with this single principle you
could suddenly describe everything from
light reflecting and refracting to The
Swinging of a pendulum clock to planets
orbiting the sun and stars orbiting the
core of the Galaxy what used to be
viewed as entirely separate fields of
physics were now all unified under a
single simple rule the variation of the
action is
zero after Oiler found out about lr's
proof he wrote to him how satisfied
would not Mr M be were he still alive if
he could see his principle of least
action apply apped to the highest degree
of dignity to which it is
[Music]
susceptible with lr's proof we now have
two ways to solve any mechanics problem
you can either use forces and vectors or
you can use energies and scalers it
seems like the principle of least action
is just way too complicated it is so
unnecessary who would ever use this you
know when you could just use Newton
Second Law like that's a piece of cake
your options are either use all of this
or just start with f equals Ma and they
give you the same answer why ever use
the principal lece action well that's
because orand L granch came up with a
way to make all of this like super super
simple if this is the action then T
minus V is called the lran now let's
replace everything we did before with
the
lran then you see that the principle of
least action works whenever this
differential equ equation is satisfied
so all you have to do now if you want to
solve any mechanics problem is you just
write down the kinetic and potential
energy and you plug it into this
equation and you're done and that
becomes extremely powerful I remember
thinking man force is like hard to get
the right answer you can do it if you're
good and people who are good at
mechanics can do it but with the lonian
approach you have this machine crank out
the principle of least action on it and
you get the right equation of motion and
you don't have to be a good physicist
that was what I took from it that as a
math guy I can do physics thanks to LR
an Oiler and it doesn't just work in one
dimension if you have more dimensions
then you just solve the oily lrange
equation for each coordinate another
thing that's great is you could use
weird coordinate systems that might be
better suited to the problem like if you
were doing a problem with something
rotating you might want to use polar
coordinates instead of cartisian
coordinates it's in will give you the
correct equations of motion in polar
coordinates which again might be kind of
tricky to do with vectors like with the
double pendulum trying to solve this
using forces is extremely hard because
as one pendulum is swinging it provides
the attachment point for the pendulum
hanging below it and so that pendulum is
in this moving reference frame as it's
swinging it's a very nasty job to write
down the correct FAL ma for a double
pendulum but if you write it down with
kinetic and potential energy it's pretty
easy this is actually how we made this
simulation now there is one little side
note we should give about the principle
of least action because the name is a
little misleading although we often
refer to the principle as the principle
of least
action it's probably good to add a
little caveat here that sometimes it's
not necessarily the minimum just as when
you find in calculus when you set a
derivative to zero that doesn't
guarantee you're getting the minimum of
a function the principle of least action
more properly stated should be the
principle of stationary action that the
laws of motion come from demanding a
stationary point which is tant amount to
this condition of setting a certain
derivative to zero and then getting the
oiler lrange equation from that so very
often it is a true minimum but but not
always but action is much more
fundamental than just classical
mechanics around the turn of the 20th
century action popped up as the key part
of a solution to one of the biggest
problems in atomic physics at the time
the UV catastrophe it's kind of spooky
that um this breakthrough that starts
the ball rolling toward quantum theory
brings action in not energy and not
Force
action uh gives you a hint yeah but that
and much more we'll have to wait for a
separate video so make sure you
subscribe to get notified when it comes
out
the story of the principle of least
action is the story of how knowledge
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