All language subtitles for Physics Students Need to Know These 5 Methods for Differential Equations [0kY3Wpvutfs].en

in this video I'm going to teach you

in this video I'm going to teach you
five methods for solving differential

five methods for solving differential

five methods for solving differential
equations that are extremely useful for

equations that are extremely useful for

equations that are extremely useful for
physics starting from the simplest and

physics starting from the simplest and

physics starting from the simplest and
working up to the most advanced but also

working up to the most advanced but also

working up to the most advanced but also
the most powerful the fact is just about

the most powerful the fact is just about

the most powerful the fact is just about
any time you want to solve a problem in

any time you want to solve a problem in

any time you want to solve a problem in
physics you're going to wind up facing a

physics you're going to wind up facing a

physics you're going to wind up facing a
differential equation in Newtonian

differential equation in Newtonian

differential equation in Newtonian
mechanics that means adding up all the

mechanics that means adding up all the

mechanics that means adding up all the
forces on an object plugging that into f

forces on an object plugging that into f

forces on an object plugging that into f
equals ma or better yet M times the

equals ma or better yet M times the

equals ma or better yet M times the
second derivative of the position and

second derivative of the position and

second derivative of the position and
then solving this differential equation

then solving this differential equation

then solving this differential equation
for the position as a function of time

for the position as a function of time

for the position as a function of time
that's not too hard for the simplest

that's not too hard for the simplest

that's not too hard for the simplest
systems we all meet in our first physics

systems we all meet in our first physics

systems we all meet in our first physics
classes but as you study more and more

classes but as you study more and more

classes but as you study more and more
physics you'll very quickly discover

physics you'll very quickly discover

physics you'll very quickly discover
that the f equals ma equation can become

that the f equals ma equation can become

that the f equals ma equation can become
extremely difficult to solve even for

extremely difficult to solve even for

extremely difficult to solve even for
setups that look like they should be

setups that look like they should be

setups that look like they should be
fairly straightforward at first glance

fairly straightforward at first glance

fairly straightforward at first glance
so it's hugely important to have a

so it's hugely important to have a

so it's hugely important to have a
toolkit of strategies for tackling the

toolkit of strategies for tackling the

toolkit of strategies for tackling the
many differential equations you're going

many differential equations you're going

many differential equations you're going
to meet throughout your physics studies

to meet throughout your physics studies

to meet throughout your physics studies
and that's why you need to learn the

and that's why you need to learn the

and that's why you need to learn the
five solution methods I'm going to tell

five solution methods I'm going to tell

five solution methods I'm going to tell
you about in this video we'll see how

you about in this video we'll see how

you about in this video we'll see how
they all work using one of the simplest

they all work using one of the simplest

they all work using one of the simplest
but also arguably the most important

but also arguably the most important

but also arguably the most important
differential equation in classical

differential equation in classical

differential equation in classical
mechanics the equation of a simple

mechanics the equation of a simple

mechanics the equation of a simple
harmonic oscillator or in other words

harmonic oscillator or in other words

harmonic oscillator or in other words
the f equals ma equation for a block

the f equals ma equation for a block

the f equals ma equation for a block
attached to a spring there's a good

attached to a spring there's a good

attached to a spring there's a good
chance you've run into this equation

chance you've run into this equation

chance you've run into this equation
before and maybe you've already seen a

before and maybe you've already seen a

before and maybe you've already seen a
couple of different ways of solving it

couple of different ways of solving it

couple of different ways of solving it
but what's hopefully going to be fun and

but what's hopefully going to be fun and

but what's hopefully going to be fun and
different about this video is that the

different about this video is that the

different about this video is that the
five solution methods I'm going to show

five solution methods I'm going to show

five solution methods I'm going to show
you will start from the most

you will start from the most

you will start from the most
straightforward and work our way up

straightforward and work our way up

straightforward and work our way up
through increasingly Advanced approaches

through increasingly Advanced approaches

through increasingly Advanced approaches
so we'll start off seeing some

so we'll start off seeing some

so we'll start off seeing some
relatively basic strategies for solving

relatively basic strategies for solving

relatively basic strategies for solving
equations like this which will already

equations like this which will already

equations like this which will already
take you a long way with lots of

take you a long way with lots of

take you a long way with lots of
problems you'll meet in classical

problems you'll meet in classical

problems you'll meet in classical
mechanics and Beyond like solving by

mechanics and Beyond like solving by

mechanics and Beyond like solving by
making a substitution or using energy

making a substitution or using energy

making a substitution or using energy
conservation but as we go along I'm

conservation but as we go along I'm

conservation but as we go along I'm
going to introduce you to some more and

going to introduce you to some more and

going to introduce you to some more and
more sophisticated techniques like using

more sophisticated techniques like using

more sophisticated techniques like using
a series expansion to solve the equation

a series expansion to solve the equation

a series expansion to solve the equation
using an integral transform like the

using an integral transform like the

using an integral transform like the
Laplace transform and finally using

Laplace transform and finally using

Laplace transform and finally using
Hamilton's equations which also give us

Hamilton's equations which also give us

Hamilton's equations which also give us
a new way of visualizing the solution as

a new way of visualizing the solution as

a new way of visualizing the solution as
what's called a flow on face space and

what's called a flow on face space and

what's called a flow on face space and
that's incredibly powerful so make sure

that's incredibly powerful so make sure

that's incredibly powerful so make sure
you stick around to the end to see that

you stick around to the end to see that

you stick around to the end to see that
okay let's get going first of all let me

okay let's get going first of all let me

okay let's get going first of all let me
quickly remind you where this

quickly remind you where this

quickly remind you where this
differential equation comes from our

differential equation comes from our

differential equation comes from our
setup is a block of mass m sitting on a

setup is a block of mass m sitting on a

setup is a block of mass m sitting on a
frictionless table and hooked up to a

frictionless table and hooked up to a

frictionless table and hooked up to a
spring of stiffness K in equilibrium the

spring of stiffness K in equilibrium the

spring of stiffness K in equilibrium the
spring isn't stretched or compressed and

spring isn't stretched or compressed and

spring isn't stretched or compressed and
the block can sit happily at rest there

the block can sit happily at rest there

the block can sit happily at rest there
let's call that position x equals zero

let's call that position x equals zero

let's call that position x equals zero
but if we Slide the block away from

but if we Slide the block away from

but if we Slide the block away from
there the spring will now exert a force

there the spring will now exert a force

there the spring will now exert a force
minus KX trying to pull the block back

minus KX trying to pull the block back

minus KX trying to pull the block back
toward equilibrium then the f equals ma

toward equilibrium then the f equals ma

toward equilibrium then the f equals ma
equation is simply M times the second

equation is simply M times the second

equation is simply M times the second
derivative of x that's the acceleration

derivative of x that's the acceleration

derivative of x that's the acceleration
equals the force minus KX now let's say

equals the force minus KX now let's say

equals the force minus KX now let's say
we pull the block out to an initial

we pull the block out to an initial

we pull the block out to an initial
position x sub zero and then release it

position x sub zero and then release it

position x sub zero and then release it
from rest the stretch spring holds the

from rest the stretch spring holds the

from rest the stretch spring holds the
block back toward equilibrium to the

block back toward equilibrium to the

block back toward equilibrium to the
left but then the block overshoots x

left but then the block overshoots x

left but then the block overshoots x
equals zero and moves to the left of

equals zero and moves to the left of

equals zero and moves to the left of
equilibrium the spring gets compressed

equilibrium the spring gets compressed

equilibrium the spring gets compressed
and pushes the block back toward the

and pushes the block back toward the

and pushes the block back toward the
right and on and on it goes making the

right and on and on it goes making the

right and on and on it goes making the
block oscillate back and forth around

block oscillate back and forth around

block oscillate back and forth around
equilibrium forever this is what we call

equilibrium forever this is what we call

equilibrium forever this is what we call
simple harmonic motion I made a separate

simple harmonic motion I made a separate

simple harmonic motion I made a separate
video All About It explaining why it's

video All About It explaining why it's

video All About It explaining why it's
arguably the most important system in

arguably the most important system in

arguably the most important system in
physics and why it shows up absolutely

physics and why it shows up absolutely

physics and why it shows up absolutely
everywhere but now let's see how to

everywhere but now let's see how to

everywhere but now let's see how to
solve for the motion from this equation

solve for the motion from this equation

solve for the motion from this equation
we're looking for X of T the position of

we're looking for X of T the position of

we're looking for X of T the position of
the block as a function of time and f

the block as a function of time and f

the block as a function of time and f
equals m a is a differential equation

equals m a is a differential equation

equals m a is a differential equation
because it involves the derivatives of

because it involves the derivatives of

because it involves the derivatives of
this function it says that the second

this function it says that the second

this function it says that the second
derivative of x with respect to T equals

derivative of x with respect to T equals

derivative of x with respect to T equals
minus K Over M times x again and in the

minus K Over M times x again and in the

minus K Over M times x again and in the
rest of this video we're going to

rest of this video we're going to

rest of this video we're going to
explore five increasingly Advanced

explore five increasingly Advanced

explore five increasingly Advanced
methods for solving this equation

methods for solving this equation

methods for solving this equation
starting off with number one it might

starting off with number one it might

starting off with number one it might
sound a little silly but honestly the

sound a little silly but honestly the

sound a little silly but honestly the
first thing you can do especially with a

first thing you can do especially with a

first thing you can do especially with a
relatively simple looking equation like

relatively simple looking equation like

relatively simple looking equation like
this one is to try to guess the solution

this one is to try to guess the solution

this one is to try to guess the solution
except that guessing doesn't sound very

except that guessing doesn't sound very

except that guessing doesn't sound very
sophisticated so instead you'll often

sophisticated so instead you'll often

sophisticated so instead you'll often
see textbooks call it making an ansats

see textbooks call it making an ansats

see textbooks call it making an ansats
which is German and sounds much fancier

which is German and sounds much fancier

which is German and sounds much fancier
all that means in

all that means in

all that means in
is we're going to ask ourselves if we

is we're going to ask ourselves if we

is we're going to ask ourselves if we
can think of a function which when we

can think of a function which when we

can think of a function which when we
take its derivative two times we get

take its derivative two times we get

take its derivative two times we get
back the same function we started with

back the same function we started with

back the same function we started with
times some negative number

times some negative number

times some negative number
so what kind of function satisfies a

so what kind of function satisfies a

so what kind of function satisfies a
property like that using our physical

property like that using our physical

property like that using our physical
intuition like we talked about before

intuition like we talked about before

intuition like we talked about before
that the block is going to oscillate

that the block is going to oscillate

that the block is going to oscillate
back and forth around equilibrium

back and forth around equilibrium

back and forth around equilibrium
functions like sine and cosine might

functions like sine and cosine might

functions like sine and cosine might
come to mind so let's make our onsets

come to mind so let's make our onsets

come to mind so let's make our onsets
and write down a guess of the form a

and write down a guess of the form a

and write down a guess of the form a
cosine Omega T where a and Omega are

cosine Omega T where a and Omega are

cosine Omega T where a and Omega are
some constants that we don't know yet

some constants that we don't know yet

some constants that we don't know yet
the idea is to see if we can choose them

the idea is to see if we can choose them

the idea is to see if we can choose them
to solve the equation we have to have

to solve the equation we have to have

to solve the equation we have to have
some constants there just to get the

some constants there just to get the

some constants there just to get the
units right X is supposed to be a length

units right X is supposed to be a length

units right X is supposed to be a length
remember in meters say that means a had

remember in meters say that means a had

remember in meters say that means a had
better have units of meters too and

better have units of meters too and

better have units of meters too and
inside the parentheses Omega T had

inside the parentheses Omega T had

inside the parentheses Omega T had
better be measured in radians which are

better be measured in radians which are

better be measured in radians which are
dimensionless so Omega had better be

dimensionless so Omega had better be

dimensionless so Omega had better be
something in radians per second in order

something in radians per second in order

something in radians per second in order
to cancel out the seconds units from the

to cancel out the seconds units from the

to cancel out the seconds units from the
T okay well let's substitute this guess

T okay well let's substitute this guess

T okay well let's substitute this guess
into the equation and see if it actually

into the equation and see if it actually

into the equation and see if it actually
works the derivative of cosine is minus

works the derivative of cosine is minus

works the derivative of cosine is minus
sine and by the chain rule we also need

sine and by the chain rule we also need

sine and by the chain rule we also need
to multiply by the derivative of the

to multiply by the derivative of the

to multiply by the derivative of the
thing in parentheses with respect to T

thing in parentheses with respect to T

thing in parentheses with respect to T
which gives us a factor of Omega

which gives us a factor of Omega

which gives us a factor of Omega
now to do it again for the second

now to do it again for the second

now to do it again for the second
derivative this time the derivative of

derivative this time the derivative of

derivative this time the derivative of
sine is cosine and again we get an extra

sine is cosine and again we get an extra

sine is cosine and again we get an extra
factor of Omega from the chain rule

factor of Omega from the chain rule

factor of Omega from the chain rule
all right that's what our guess gives us

all right that's what our guess gives us

all right that's what our guess gives us
for the second derivative but does it

for the second derivative but does it

for the second derivative but does it
solve our differential equation it looks

solve our differential equation it looks

solve our differential equation it looks
promising because it says that the

promising because it says that the

promising because it says that the
second derivative of x is indeed equal

second derivative of x is indeed equal

second derivative of x is indeed equal
to X again times a constant minus Omega

to X again times a constant minus Omega

to X again times a constant minus Omega
squared and all we need to do is pick

squared and all we need to do is pick

squared and all we need to do is pick
this number Omega squared to be the same

this number Omega squared to be the same

this number Omega squared to be the same
as the ratio K Over M that appeared in

as the ratio K Over M that appeared in

as the ratio K Over M that appeared in
the differential equation and that'll do

the differential equation and that'll do

the differential equation and that'll do
it if we choose this value for Omega

it if we choose this value for Omega

it if we choose this value for Omega
then X of T equals a cosine Omega T will

then X of T equals a cosine Omega T will

then X of T equals a cosine Omega T will
indeed satisfy the equation

indeed satisfy the equation

indeed satisfy the equation
so are we done well no first of all sine

so are we done well no first of all sine

so are we done well no first of all sine
of Omega T satisfies this property just

of Omega T satisfies this property just

of Omega T satisfies this property just
as well as cosine Omega T and so more

as well as cosine Omega T and so more

as well as cosine Omega T and so more
generally we can add them together to

generally we can add them together to

generally we can add them together to
write a general solution of this form

write a general solution of this form

write a general solution of this form
that works because the differential

that works because the differential

that works because the differential
equation is linear meaning that we only

equation is linear meaning that we only

equation is linear meaning that we only
have single powers of X and its

have single powers of X and its

have single powers of X and its
derivatives showing up but what are we

derivatives showing up but what are we

derivatives showing up but what are we
supposed to do with these two constants

supposed to do with these two constants

supposed to do with these two constants
A and B this expression solves the

A and B this expression solves the

A and B this expression solves the
equation for any values of these numbers

equation for any values of these numbers

equation for any values of these numbers
that brings us to a really important

that brings us to a really important

that brings us to a really important
point about solving differential

point about solving differential

point about solving differential
equations the equation itself is only

equations the equation itself is only

equations the equation itself is only
half the story we also have to specify

half the story we also have to specify

half the story we also have to specify
the initial conditions we want to

the initial conditions we want to

the initial conditions we want to
satisfy in order to get the solution to

satisfy in order to get the solution to

satisfy in order to get the solution to
the problem physically that makes total

the problem physically that makes total

the problem physically that makes total
sense when you throw a ball up into the

sense when you throw a ball up into the

sense when you throw a ball up into the
air we need to know the initial position

air we need to know the initial position

air we need to know the initial position
you're throwing it from and the initial

you're throwing it from and the initial

you're throwing it from and the initial
velocity in order to be able to say what

velocity in order to be able to say what

velocity in order to be able to say what
trajectory it's going to follow

trajectory it's going to follow

trajectory it's going to follow
likewise we need to know the initial

likewise we need to know the initial

likewise we need to know the initial
position and initial velocity of the

position and initial velocity of the

position and initial velocity of the
Block in order to say what its Position

Block in order to say what its Position

Block in order to say what its Position
will be after that in this case we

will be after that in this case we

will be after that in this case we
release the block from rest at x sub

release the block from rest at x sub

release the block from rest at x sub
zero and that means our two initial

zero and that means our two initial

zero and that means our two initial
conditions are these

conditions are these

conditions are these
mathematically the fact that we need two

mathematically the fact that we need two

mathematically the fact that we need two
initial conditions comes from the fact

initial conditions comes from the fact

initial conditions comes from the fact
that the differential equation is second

that the differential equation is second

that the differential equation is second
order meaning that the highest

order meaning that the highest

order meaning that the highest
derivative that shows up is the second

derivative that shows up is the second

derivative that shows up is the second
derivative of x

derivative of x

derivative of x
so when we plug in t equals zero the

so when we plug in t equals zero the

so when we plug in t equals zero the
sign disappears and cosine of 0 is equal

sign disappears and cosine of 0 is equal

sign disappears and cosine of 0 is equal
to one so we'd better set a equal to X

to one so we'd better set a equal to X

to one so we'd better set a equal to X
Sub 0 in order to solve our specific

Sub 0 in order to solve our specific

Sub 0 in order to solve our specific
problem likewise if you take the

problem likewise if you take the

problem likewise if you take the
derivative and demand at the initial

derivative and demand at the initial

derivative and demand at the initial
velocity vanishes you'll see that we

velocity vanishes you'll see that we

velocity vanishes you'll see that we
need to set b equal to zero and that

need to set b equal to zero and that

need to set b equal to zero and that
leaves us with X of T equals x Sub 0

leaves us with X of T equals x Sub 0

leaves us with X of T equals x Sub 0
cosine Omega T where again Omega equals

cosine Omega T where again Omega equals

cosine Omega T where again Omega equals
the square root of K Over m is fixed by

the square root of K Over m is fixed by

the square root of K Over m is fixed by
the stiffness of the spring and the mass

the stiffness of the spring and the mass

the stiffness of the spring and the mass
of the block

of the block

of the block
this looks about like we'd expect the

this looks about like we'd expect the

this looks about like we'd expect the
block starts out at rest at the initial

block starts out at rest at the initial

block starts out at rest at the initial
displacement x sub zero and then when we

displacement x sub zero and then when we

displacement x sub zero and then when we
let it go it oscillates back and forth

let it go it oscillates back and forth

let it go it oscillates back and forth
around equilibrium where Omega controls

around equilibrium where Omega controls

around equilibrium where Omega controls
how fast it oscillates so there we have

how fast it oscillates so there we have

how fast it oscillates so there we have
it we've solved the differential

it we've solved the differential

it we've solved the differential
equation together with the initial

equation together with the initial

equation together with the initial
conditions by substituting in a guess or

conditions by substituting in a guess or

conditions by substituting in a guess or
onsots with some constants in it and

onsots with some constants in it and

onsots with some constants in it and
seeing how to pick the constants in

seeing how to pick the constants in

seeing how to pick the constants in
order to get a solution and this kind of

order to get a solution and this kind of

order to get a solution and this kind of
strategy Works in general for a linear

strategy Works in general for a linear

strategy Works in general for a linear
equation like this where a B and C are

equation like this where a B and C are

equation like this where a B and C are
some consonants in general you'd pick an

some consonants in general you'd pick an

some consonants in general you'd pick an
exponential for your guess a e to the

exponential for your guess a e to the

exponential for your guess a e to the
Omega T substituted in and see what

Omega T substituted in and see what

Omega T substituted in and see what
conditions come out on those constants

conditions come out on those constants

conditions come out on those constants
explaining that whole method in detail

explaining that whole method in detail

explaining that whole method in detail
though would really deserve its own

though would really deserve its own

though would really deserve its own
video

video

video
for now we're going to stick with the

for now we're going to stick with the

for now we're going to stick with the
simple harmonic oscillator equation and

simple harmonic oscillator equation and

simple harmonic oscillator equation and
see four other really powerful ways of

see four other really powerful ways of

see four other really powerful ways of
approaching it and that brings us to

approaching it and that brings us to

approaching it and that brings us to
Method number two using energy

Method number two using energy

Method number two using energy
conservation to solve the equation as

conservation to solve the equation as

conservation to solve the equation as
you've hopefully learned before if we

you've hopefully learned before if we

you've hopefully learned before if we
take the kinetic energy of the block one

take the kinetic energy of the block one

take the kinetic energy of the block one
half M times the velocity squared and

half M times the velocity squared and

half M times the velocity squared and
add to it the potential energy in the

add to it the potential energy in the

add to it the potential energy in the
spring one-half k x squared will get a

spring one-half k x squared will get a

spring one-half k x squared will get a
constant the total energy that's not

constant the total energy that's not

constant the total energy that's not
obvious because both X and DX by DT are

obvious because both X and DX by DT are

obvious because both X and DX by DT are
changing with time as the block slides

changing with time as the block slides

changing with time as the block slides
back and forth but when we add them

back and forth but when we add them

back and forth but when we add them
together in this special combination the

together in this special combination the

together in this special combination the
t's drop out and we get a constant the

t's drop out and we get a constant the

t's drop out and we get a constant the
way to check that that's true is to take

way to check that that's true is to take

way to check that that's true is to take
the derivative of e with respect to T

the derivative of e with respect to T

the derivative of e with respect to T
and see that it's equal to zero I'll

and see that it's equal to zero I'll

and see that it's equal to zero I'll
show you how to prove that in the notes

show you how to prove that in the notes

show you how to prove that in the notes
which you can get at the link in the

which you can get at the link in the

which you can get at the link in the
description and where I'll go through

description and where I'll go through

description and where I'll go through
everything we're covering in this video

everything we're covering in this video

everything we're covering in this video
in more detail if you really want to

in more detail if you really want to

in more detail if you really want to
learn all these Concepts you should

learn all these Concepts you should

learn all these Concepts you should
watch first to get the general idea of

watch first to get the general idea of

watch first to get the general idea of
how things work and and then go through

how things work and and then go through

how things work and and then go through
the notes to take your time processing

the notes to take your time processing

the notes to take your time processing
the material in this case the potential

the material in this case the potential

the material in this case the potential
energy is a parabola

energy is a parabola

energy is a parabola
so when we release the block somewhere

so when we release the block somewhere

so when we release the block somewhere
over here at x sub zero all of the

over here at x sub zero all of the

over here at x sub zero all of the
energy is the potential energy stored in

energy is the potential energy stored in

energy is the potential energy stored in
the spring there's no kinetic energy

the spring there's no kinetic energy

the spring there's no kinetic energy
because we're releasing the block from

because we're releasing the block from

because we're releasing the block from
rest then when we let it go the block

rest then when we let it go the block

rest then when we let it go the block
starts to speed up and the spring starts

starts to speed up and the spring starts

starts to speed up and the spring starts
to relax by the time it reaches x equals

to relax by the time it reaches x equals

to relax by the time it reaches x equals
zero all the energy is kinetic and on

zero all the energy is kinetic and on

zero all the energy is kinetic and on
and on the energy Cycles back and forth

and on the energy Cycles back and forth

and on the energy Cycles back and forth
between kinetic and potential but the

between kinetic and potential but the

between kinetic and potential but the
total energy never changes it's always

total energy never changes it's always

total energy never changes it's always
the same number we started with one half

the same number we started with one half

the same number we started with one half
k x sub zero squared and what we'll see

k x sub zero squared and what we'll see

k x sub zero squared and what we'll see
now is that we can use this equation for

now is that we can use this equation for

now is that we can use this equation for
energy conservation to solve for the

energy conservation to solve for the

energy conservation to solve for the
trajectory of the Block it's again a

trajectory of the Block it's again a

trajectory of the Block it's again a
differential equation for x but notice

differential equation for x but notice

differential equation for x but notice
that it only involves the first

that it only involves the first

that it only involves the first
derivative of x not the second

derivative of x not the second

derivative of x not the second
derivative that we had in f equals Ma

derivative that we had in f equals Ma

derivative that we had in f equals Ma
let's rearrange the equation a bit we

let's rearrange the equation a bit we

let's rearrange the equation a bit we
can cross out the halves and I'll also

can cross out the halves and I'll also

can cross out the halves and I'll also
move the KX squared over to the left

move the KX squared over to the left

move the KX squared over to the left
hand side and then divide by m

hand side and then divide by m

hand side and then divide by m
I'll also use the same symbol Omega

I'll also use the same symbol Omega

I'll also use the same symbol Omega
squared as before for the ratio K Over M

squared as before for the ratio K Over M

squared as before for the ratio K Over M
remember Omega was what told us how fast

remember Omega was what told us how fast

remember Omega was what told us how fast
the block would oscillate back and forth

the block would oscillate back and forth

the block would oscillate back and forth
and finally we can take the square root

and finally we can take the square root

and finally we can take the square root
to get an equation for DX by DT

to get an equation for DX by DT

to get an equation for DX by DT
now something really special has

now something really special has

now something really special has
happened this equation tells us the

happened this equation tells us the

happened this equation tells us the
velocity of the block as a function of

velocity of the block as a function of

velocity of the block as a function of
its position X the point is if we know

its position X the point is if we know

its position X the point is if we know
the position of the block we know how

the position of the block we know how

the position of the block we know how
stretched or compressed the spring is

stretched or compressed the spring is

stretched or compressed the spring is
and therefore how much potential energy

and therefore how much potential energy

and therefore how much potential energy
is stored in it then conservation of the

is stored in it then conservation of the

is stored in it then conservation of the
total energy tells us how much is left

total energy tells us how much is left

total energy tells us how much is left
over for the kinetic energy of the block

over for the kinetic energy of the block

over for the kinetic energy of the block
and therefore how fast it's moving

and therefore how fast it's moving

and therefore how fast it's moving
so when the block starts off at X Sub 0

so when the block starts off at X Sub 0

so when the block starts off at X Sub 0
we get V equals zero because we released

we get V equals zero because we released

we get V equals zero because we released
it from rest but by the time the spring

it from rest but by the time the spring

it from rest but by the time the spring
pulls the block back to equilibrium at x

pulls the block back to equilibrium at x

pulls the block back to equilibrium at x
equals zero it's sped up to its maximum

equals zero it's sped up to its maximum

equals zero it's sped up to its maximum
velocity and we get DX by DT equals

velocity and we get DX by DT equals

velocity and we get DX by DT equals
Omega times x0. actually we should

Omega times x0. actually we should

Omega times x0. actually we should
really get minus that because the block

really get minus that because the block

really get minus that because the block
is initially moving to the left so we

is initially moving to the left so we

is initially moving to the left so we
ought to be a little more careful when

ought to be a little more careful when

ought to be a little more careful when
we take the square root since we can get

we take the square root since we can get

we take the square root since we can get
either sine we take the minus sign when

either sine we take the minus sign when

either sine we take the minus sign when
the block is moving to the left and the

the block is moving to the left and the

the block is moving to the left and the
plus sign when it turns around and goes

plus sign when it turns around and goes

plus sign when it turns around and goes
back to the right

back to the right

back to the right
and now we can solve for x of T by

and now we can solve for x of T by

and now we can solve for x of T by
integrating one more time just divide

integrating one more time just divide

integrating one more time just divide
the square root over to the left hand

the square root over to the left hand

the square root over to the left hand
side and multiply the DT over to the

side and multiply the DT over to the

side and multiply the DT over to the
right in order to separate the variables

right in order to separate the variables

right in order to separate the variables
next we integrate both sides of this

next we integrate both sides of this

next we integrate both sides of this
equation the integral on the right is

equation the integral on the right is

equation the integral on the right is
super easy we just get T maybe plus some

super easy we just get T maybe plus some

super easy we just get T maybe plus some
integration constant C the integral over

integration constant C the integral over

integration constant C the integral over
X is a little harder you can do it with

X is a little harder you can do it with

X is a little harder you can do it with
a trig substitution or of course you can

a trig substitution or of course you can

a trig substitution or of course you can
just look it up it's given by minus the

just look it up it's given by minus the

just look it up it's given by minus the
inverse cosine of x over x0 we could

inverse cosine of x over x0 we could

inverse cosine of x over x0 we could
also add another integration constant

also add another integration constant

also add another integration constant
here but we can just absorb that into

here but we can just absorb that into

here but we can just absorb that into
the other constant C on the right

the other constant C on the right

the other constant C on the right
now we solve for x flip the sign take

now we solve for x flip the sign take

now we solve for x flip the sign take
the cosine of both sides and move the x0

the cosine of both sides and move the x0

the cosine of both sides and move the x0
over to the right

over to the right

over to the right
now we're almost there cosine doesn't

now we're almost there cosine doesn't

now we're almost there cosine doesn't
care if you plug in plus or minus

care if you plug in plus or minus

care if you plug in plus or minus
something it's an even function so we

something it's an even function so we

something it's an even function so we
can throw out the plus or minus and as

can throw out the plus or minus and as

can throw out the plus or minus and as
for the C remember that when we plug in

for the C remember that when we plug in

for the C remember that when we plug in
t equals 0 we want to get x sub 0. so we

t equals 0 we want to get x sub 0. so we

t equals 0 we want to get x sub 0. so we
can just set c equal to zero

can just set c equal to zero

can just set c equal to zero
then at last we get X of T equals x Sub

then at last we get X of T equals x Sub

then at last we get X of T equals x Sub
0 cosine Omega T just like we found with

0 cosine Omega T just like we found with

0 cosine Omega T just like we found with
method number one so conservation of

method number one so conservation of

method number one so conservation of
energy also lets us easily get to the

energy also lets us easily get to the

energy also lets us easily get to the
solution of our differential equation

solution of our differential equation

solution of our differential equation
and in fact this strategy can often be

and in fact this strategy can often be

and in fact this strategy can often be
successful for harder problems even when

successful for harder problems even when

successful for harder problems even when
our first method wouldn't work a great

our first method wouldn't work a great

our first method wouldn't work a great
example is the simple pendulum which is

example is the simple pendulum which is

example is the simple pendulum which is
supposed to be so simple that it's in

supposed to be so simple that it's in

supposed to be so simple that it's in
the name but actually it's surprisingly

the name but actually it's surprisingly

the name but actually it's surprisingly
tricky I'll let you play with that one

tricky I'll let you play with that one

tricky I'll let you play with that one
for yourself for practice with this

for yourself for practice with this

for yourself for practice with this
method and I'll share some more details

method and I'll share some more details

method and I'll share some more details
about it in the notes

about it in the notes

about it in the notes
so now we've seen two different ways of

so now we've seen two different ways of

so now we've seen two different ways of
solving the harmonic oscillator equation

solving the harmonic oscillator equation

solving the harmonic oscillator equation
and these will more or less do the job

and these will more or less do the job

and these will more or less do the job
for most of the equations you'll meet in

for most of the equations you'll meet in

for most of the equations you'll meet in
your first mechanics class but if you're

your first mechanics class but if you're

your first mechanics class but if you're
up for it what I'd like to do now is

up for it what I'd like to do now is

up for it what I'd like to do now is
show you some more powerful methods that

show you some more powerful methods that

show you some more powerful methods that
will come in handy later on when you're

will come in handy later on when you're

will come in handy later on when you're
faced with harder equations so let's

faced with harder equations so let's

faced with harder equations so let's
plow ahead to Method number three using

plow ahead to Method number three using

plow ahead to Method number three using
a series expansion this one is probably

a series expansion this one is probably

a series expansion this one is probably
the most versatile of all the strategies

the most versatile of all the strategies

the most versatile of all the strategies
we'll see here and you can apply it to

we'll see here and you can apply it to

we'll see here and you can apply it to
most any differential equation to get an

most any differential equation to get an

most any differential equation to get an
exact or even just an approximate

exact or even just an approximate

exact or even just an approximate
solution the idea is whatever the

solution the idea is whatever the

solution the idea is whatever the
solution X of T to our differential

solution X of T to our differential

solution X of T to our differential
equation might be we can almost always

equation might be we can almost always

equation might be we can almost always
expand it as a Taylor series in powers

expand it as a Taylor series in powers

expand it as a Taylor series in powers
of T at least within a window where

of T at least within a window where

of T at least within a window where
things are well behaved the question is

things are well behaved the question is

things are well behaved the question is
how do we figure out what these

how do we figure out what these

how do we figure out what these
coefficients are supposed to be

coefficients are supposed to be

coefficients are supposed to be
well first of all let's go ahead and

well first of all let's go ahead and

well first of all let's go ahead and
impose our initial conditions

impose our initial conditions

impose our initial conditions
when we plug in t equals 0 to the series

when we plug in t equals 0 to the series

when we plug in t equals 0 to the series
expansion all the t's disappear and

expansion all the t's disappear and

expansion all the t's disappear and
we're left with X of 0 equals a sub zero

we're left with X of 0 equals a sub zero

we're left with X of 0 equals a sub zero
so we want to set that equal to x0 to

so we want to set that equal to x0 to

so we want to set that equal to x0 to
coincide with the initial position of

coincide with the initial position of

coincide with the initial position of
the block

the block

the block
and to impose that the initial velocity

and to impose that the initial velocity

and to impose that the initial velocity
is zero we'll take the derivative of the

is zero we'll take the derivative of the

is zero we'll take the derivative of the
series a one plus two a two times t plus

series a one plus two a two times t plus

series a one plus two a two times t plus
three a three t squared and so on

three a three t squared and so on

three a three t squared and so on
now when we plug in t equals zero we're

now when we plug in t equals zero we're

now when we plug in t equals zero we're
left with a sub 1. and so we want to set

left with a sub 1. and so we want to set

left with a sub 1. and so we want to set
that equal to zero

that equal to zero

that equal to zero
all right so far we figured out a Sub 0

all right so far we figured out a Sub 0

all right so far we figured out a Sub 0
and a sub 1 but there are still

and a sub 1 but there are still

and a sub 1 but there are still
infinitely many coefficients left to

infinitely many coefficients left to

infinitely many coefficients left to
determine so the next thing we need to

determine so the next thing we need to

determine so the next thing we need to
do is actually plug the expansion into

do is actually plug the expansion into

do is actually plug the expansion into
the differential equation that means we

the differential equation that means we

the differential equation that means we
need to take the derivative of the

need to take the derivative of the

need to take the derivative of the
series one more time to get the

series one more time to get the

series one more time to get the
acceleration we'll have 2 times A2 plus

acceleration we'll have 2 times A2 plus

acceleration we'll have 2 times A2 plus
3 times 2 a 3T plus four times three a

3 times 2 a 3T plus four times three a

3 times 2 a 3T plus four times three a
four t squared and so on

four t squared and so on

four t squared and so on
and now we add on Omega squared times x

and now we add on Omega squared times x

and now we add on Omega squared times x
and set the whole thing equal to zero

and set the whole thing equal to zero

and set the whole thing equal to zero
and it's helpful to pair up the

and it's helpful to pair up the

and it's helpful to pair up the
corresponding terms

all of this needs to vanish if we want

all of this needs to vanish if we want
our series to solve the differential

our series to solve the differential

our series to solve the differential
equation and the only way that can

equation and the only way that can

equation and the only way that can
happen for every time T is if all the

happen for every time T is if all the

happen for every time T is if all the
coefficients are separately equal to

coefficients are separately equal to

coefficients are separately equal to
zero so the idea is to go term by term

zero so the idea is to go term by term

zero so the idea is to go term by term
through the series and demand that each

through the series and demand that each

through the series and demand that each
factor in parentheses is zero let's

factor in parentheses is zero let's

factor in parentheses is zero let's
start with the odd terms the coefficient

start with the odd terms the coefficient

start with the odd terms the coefficient
of the T term is 3 times 2 a sub 3. so

of the T term is 3 times 2 a sub 3. so

of the T term is 3 times 2 a sub 3. so
for that to vanish we need to choose A3

for that to vanish we need to choose A3

for that to vanish we need to choose A3
equal to zero notice there's also an A3

equal to zero notice there's also an A3

equal to zero notice there's also an A3
in the T Cube term so I'll go ahead and

in the T Cube term so I'll go ahead and

in the T Cube term so I'll go ahead and
erase that as well

erase that as well

erase that as well
but now when we look at that t cubed

but now when we look at that t cubed

but now when we look at that t cubed
term its coefficient is just 5 times 4 a

term its coefficient is just 5 times 4 a

term its coefficient is just 5 times 4 a
sub 5. and so for that to vanish we'll

sub 5. and so for that to vanish we'll

sub 5. and so for that to vanish we'll
also have to set a 5 equal to zero

also have to set a 5 equal to zero

also have to set a 5 equal to zero
the same thing is going to happen for

the same thing is going to happen for

the same thing is going to happen for
all the odd terms so we conclude that

all the odd terms so we conclude that

all the odd terms so we conclude that
all the odd coefficients are equal to

all the odd coefficients are equal to

all the odd coefficients are equal to
zero that's already pretty nice because

zero that's already pretty nice because

zero that's already pretty nice because
it means we get to throw out half the

it means we get to throw out half the

it means we get to throw out half the
terms in our expansion so now let's move

terms in our expansion so now let's move

terms in our expansion so now let's move
on to the even terms the zeroth one says

on to the even terms the zeroth one says

on to the even terms the zeroth one says
that 2 a 2 plus Omega squared x0 is

that 2 a 2 plus Omega squared x0 is

that 2 a 2 plus Omega squared x0 is
equal to zero and so we can solve that

equal to zero and so we can solve that

equal to zero and so we can solve that
for a sub 2. next for the t-squared term

for a sub 2. next for the t-squared term

for a sub 2. next for the t-squared term
we've got 4 times 3 a 4 plus Omega

we've got 4 times 3 a 4 plus Omega

we've got 4 times 3 a 4 plus Omega
squared A2 and we set that equal to zero

squared A2 and we set that equal to zero

squared A2 and we set that equal to zero
again we can solve to get a sub 4.

again we can solve to get a sub 4.

again we can solve to get a sub 4.
don't worry too much about that algebra

don't worry too much about that algebra

don't worry too much about that algebra
the point is you can already see the

the point is you can already see the

the point is you can already see the
pattern that's forming here here are the

pattern that's forming here here are the

pattern that's forming here here are the
first few terms we're getting for our

first few terms we're getting for our

first few terms we're getting for our
series solution does that look familiar

series solution does that look familiar

series solution does that look familiar
at all let's simplify it a bit by

at all let's simplify it a bit by

at all let's simplify it a bit by
pulling out the common factor of x0 and

pulling out the common factor of x0 and

pulling out the common factor of x0 and
we can also put the omegas in the t's

we can also put the omegas in the t's

we can also put the omegas in the t's
together like this

together like this

together like this
so how about now does this thing look

so how about now does this thing look

so how about now does this thing look
like the Taylor series for any function

like the Taylor series for any function

like the Taylor series for any function
that you know that's right the sum in

that you know that's right the sum in

that you know that's right the sum in
parentheses is just the Taylor series

parentheses is just the Taylor series

parentheses is just the Taylor series
for the cosine and so reassuringly we've

for the cosine and so reassuringly we've

for the cosine and so reassuringly we've
once again found that X of T equals x0

once again found that X of T equals x0

once again found that X of T equals x0
cosine Omega t

cosine Omega t

cosine Omega t
like I mentioned series expansions like

like I mentioned series expansions like

like I mentioned series expansions like
this are an extremely versatile method

this are an extremely versatile method

this are an extremely versatile method
for solving all kinds of differential

for solving all kinds of differential

for solving all kinds of differential
equations they don't always add up to a

equations they don't always add up to a

equations they don't always add up to a
simple looking function like this but

simple looking function like this but

simple looking function like this but
that doesn't make them any less useful

that doesn't make them any less useful

that doesn't make them any less useful
or valid as a solution to the equation

or valid as a solution to the equation

or valid as a solution to the equation
as long as you're looking at a point

as long as you're looking at a point

as long as you're looking at a point
where the series converges

where the series converges

where the series converges
okay we're on a roll here let's keep it

okay we're on a roll here let's keep it

okay we're on a roll here let's keep it
going with our next method using an

going with our next method using an

going with our next method using an
integral transform to solve a

integral transform to solve a

integral transform to solve a
differential equation we're definitely

differential equation we're definitely

differential equation we're definitely
getting a little more advanced here but

getting a little more advanced here but

getting a little more advanced here but
this is really cool so stick with me

this is really cool so stick with me

this is really cool so stick with me
there are lots of kinds of integral

there are lots of kinds of integral

there are lots of kinds of integral
transforms out there including the

transforms out there including the

transforms out there including the
Fourier transform which my last video

Fourier transform which my last video

Fourier transform which my last video
was actually all about but the one

was actually all about but the one

was actually all about but the one
that's most useful for solving the

that's most useful for solving the

that's most useful for solving the
problem we're looking at today is called

problem we're looking at today is called

problem we're looking at today is called
the Laplace transform and here's what it

the Laplace transform and here's what it

the Laplace transform and here's what it
is the Laplace transform is an

is the Laplace transform is an

is the Laplace transform is an
instruction to take our position

instruction to take our position

instruction to take our position
function X of T multiply it by e to the

function X of T multiply it by e to the

function X of T multiply it by e to the
minus St with some new variable called s

minus St with some new variable called s

minus St with some new variable called s
and then integrate that over T from 0

and then integrate that over T from 0

and then integrate that over T from 0
all the way to T equals infinity and

all the way to T equals infinity and

all the way to T equals infinity and
we'll call that X hat of s

we'll call that X hat of s

we'll call that X hat of s
okay well that sounds like a funny thing

okay well that sounds like a funny thing

okay well that sounds like a funny thing
to do especially if you've never seen it

to do especially if you've never seen it

to do especially if you've never seen it
before but we'll see in a second that

before but we'll see in a second that

before but we'll see in a second that
this transformation has a magical

this transformation has a magical

this transformation has a magical
property when it comes to differential

property when it comes to differential

property when it comes to differential
equations the way you should think about

equations the way you should think about

equations the way you should think about
it though is that we have two spaces

it though is that we have two spaces

it though is that we have two spaces
here t-space where our original function

here t-space where our original function

here t-space where our original function
X of T lives and s space where it's

X of T lives and s space where it's

X of T lives and s space where it's
Laplace transform lives X hat of s to

Laplace transform lives X hat of s to

Laplace transform lives X hat of s to
give a couple of examples if x of T were

give a couple of examples if x of T were

give a couple of examples if x of T were
a constant like say x of T equals one

a constant like say x of T equals one

a constant like say x of T equals one
then it's just a horizontal line in t

then it's just a horizontal line in t

then it's just a horizontal line in t
space and you can show pretty easily

space and you can show pretty easily

space and you can show pretty easily
that it's Laplace transform in s space

that it's Laplace transform in s space

that it's Laplace transform in s space
obtained by doing this integral is one

obtained by doing this integral is one

obtained by doing this integral is one
over s

over s

over s
or for our block on a spring we found

or for our block on a spring we found

or for our block on a spring we found
and we're about to find again that X of

and we're about to find again that X of

and we're about to find again that X of
T equals x0 cosine Omega T it oscillates

T equals x0 cosine Omega T it oscillates

T equals x0 cosine Omega T it oscillates
in t space

in t space

in t space
and in s space it's Laplace transform is

and in s space it's Laplace transform is

and in s space it's Laplace transform is
a rational function x0 times s divided

a rational function x0 times s divided

a rational function x0 times s divided
by S squared plus Omega squared

by S squared plus Omega squared

by S squared plus Omega squared
all right so we can do this integral to

all right so we can do this integral to

all right so we can do this integral to
go from t space to s space big whoop why

go from t space to s space big whoop why

go from t space to s space big whoop why
the heck would we want to do such a

the heck would we want to do such a

the heck would we want to do such a
thing how does it help us solve a

thing how does it help us solve a

thing how does it help us solve a
differential equation like a harmonic

differential equation like a harmonic

differential equation like a harmonic
oscillator the reason is that the

oscillator the reason is that the

oscillator the reason is that the
Laplace transform acts in a beautifully

Laplace transform acts in a beautifully

Laplace transform acts in a beautifully
simple way on derivatives when we take

simple way on derivatives when we take

simple way on derivatives when we take
the transform of the derivative DX by DT

the transform of the derivative DX by DT

the transform of the derivative DX by DT
it turns into s times x hat of s minus

it turns into s times x hat of s minus

it turns into s times x hat of s minus
the initial position

the initial position

the initial position
in other words taking a derivative in t

in other words taking a derivative in t

in other words taking a derivative in t
space is the same as simply multiplying

space is the same as simply multiplying

space is the same as simply multiplying
by s in s space up to a shift by x0 and

by s in s space up to a shift by x0 and

by s in s space up to a shift by x0 and
that follows just by using integration

that follows just by using integration

that follows just by using integration
by parts in the Laplace transform

by parts in the Laplace transform

by parts in the Laplace transform
integral I'll show you how that works in

integral I'll show you how that works in

integral I'll show you how that works in
the notes but the point is because of

the notes but the point is because of

the notes but the point is because of
this beautiful property the Laplace

this beautiful property the Laplace

this beautiful property the Laplace
transform can turn a differential

transform can turn a differential

transform can turn a differential
equation for x of T into an algebraic

equation for x of T into an algebraic

equation for x of T into an algebraic
equation for x hat of s

equation for x hat of s

equation for x hat of s
let's see how that works for our

let's see how that works for our

let's see how that works for our
harmonic oscillator equation we'll take

harmonic oscillator equation we'll take

harmonic oscillator equation we'll take
the Laplace transform of both sides on

the Laplace transform of both sides on

the Laplace transform of both sides on
the right we just get that constant

the right we just get that constant

the right we just get that constant
minus Omega squared times the Laplace

minus Omega squared times the Laplace

minus Omega squared times the Laplace
transform of X on the left we need to

transform of X on the left we need to

transform of X on the left we need to
use our derivative rule twice in a row

use our derivative rule twice in a row

use our derivative rule twice in a row
if you work that out you'll get S

if you work that out you'll get S

if you work that out you'll get S
squared times x hat minus s times the

squared times x hat minus s times the

squared times x hat minus s times the
initial position minus the initial

initial position minus the initial

initial position minus the initial
velocity and if we plug in our initial

velocity and if we plug in our initial

velocity and if we plug in our initial
conditions

conditions

conditions
we find that when we transform our

we find that when we transform our

we find that when we transform our
differential equation to s space it

differential equation to s space it

differential equation to s space it
becomes s squared times x hat minus s x

becomes s squared times x hat minus s x

becomes s squared times x hat minus s x
0 equals minus Omega squared x hat like

0 equals minus Omega squared x hat like

0 equals minus Omega squared x hat like
I promised there are no more derivatives

I promised there are no more derivatives

I promised there are no more derivatives
the Laplace transform took our

the Laplace transform took our

the Laplace transform took our
differential equation for x of T and

differential equation for x of T and

differential equation for x of T and
turned it into an algebraic equation for

turned it into an algebraic equation for

turned it into an algebraic equation for
x hat of s and this equation is much

x hat of s and this equation is much

x hat of s and this equation is much
easier to solve just move the X hats

easier to solve just move the X hats

easier to solve just move the X hats
over to the left

over to the left

over to the left
and then divide out that factor out

and then divide out that factor out

and then divide out that factor out
front to get X hat of s equals s x 0

front to get X hat of s equals s x 0

front to get X hat of s equals s x 0
divided by S squared plus Omega squared

divided by S squared plus Omega squared

divided by S squared plus Omega squared
and that's the solution to our problem

and that's the solution to our problem

and that's the solution to our problem
in s space anyway

in s space anyway

in s space anyway
to finish the job we just need to

to finish the job we just need to

to finish the job we just need to
transform back to t space there's a

transform back to t space there's a

transform back to t space there's a
general formula for doing that but in

general formula for doing that but in

general formula for doing that but in
practice it's often faster to just pull

practice it's often faster to just pull

practice it's often faster to just pull
up a table of Laplace transforms there's

up a table of Laplace transforms there's

up a table of Laplace transforms there's
a nice one on Wikipedia and find the one

a nice one on Wikipedia and find the one

a nice one on Wikipedia and find the one
you're looking for in fact I already

you're looking for in fact I already

you're looking for in fact I already
mentioned that this function is the

mentioned that this function is the

mentioned that this function is the
Laplace transform of x0 cosine Omega T

Laplace transform of x0 cosine Omega T

Laplace transform of x0 cosine Omega T
and therefore that's the solution to our

and therefore that's the solution to our

and therefore that's the solution to our
original equation once again so that's

original equation once again so that's

original equation once again so that's
method number four starting from a

method number four starting from a

method number four starting from a
linear differential equation take the

linear differential equation take the

linear differential equation take the
Laplace transform to try to turn it into

Laplace transform to try to turn it into

Laplace transform to try to turn it into
an algebraic equation which you can

an algebraic equation which you can

an algebraic equation which you can
solve for x hat and finally transform

solve for x hat and finally transform

solve for x hat and finally transform
back to get your solution for x

back to get your solution for x

back to get your solution for x
all right we're in the home stretch now

all right we're in the home stretch now

all right we're in the home stretch now
and I saved maybe the most fascinating

and I saved maybe the most fascinating

and I saved maybe the most fascinating
of all of these for last Hamilton's

of all of these for last Hamilton's

of all of these for last Hamilton's
equations and flows on face space let me

equations and flows on face space let me

equations and flows on face space let me
show you how it works

show you how it works

show you how it works
we started out with the f equals m a

we started out with the f equals m a

we started out with the f equals m a
equation for a block on Spring

equation for a block on Spring

equation for a block on Spring
now notice that the left hand side is

now notice that the left hand side is

now notice that the left hand side is
the same as the derivative of M times DX

the same as the derivative of M times DX

the same as the derivative of M times DX
by DT since m is a constant in other

by DT since m is a constant in other

by DT since m is a constant in other
words it's the derivative of the

words it's the derivative of the

words it's the derivative of the
momentum p

momentum p

momentum p
that's just Newton's Second Law the

that's just Newton's Second Law the

that's just Newton's Second Law the
force minus KX equals the rate of change

force minus KX equals the rate of change

force minus KX equals the rate of change
of the momentum but mathematically what

of the momentum but mathematically what

of the momentum but mathematically what
that enables us to do is replace the

that enables us to do is replace the

that enables us to do is replace the
single second order differential

single second order differential

single second order differential
equation that we started with with a

equation that we started with with a

equation that we started with with a
pair of first order equations

pair of first order equations

pair of first order equations
these are called Hamilton's equations I

these are called Hamilton's equations I

these are called Hamilton's equations I
haven't done anything fancy this pair of

haven't done anything fancy this pair of

haven't done anything fancy this pair of
equations contains the exact same

equations contains the exact same

equations contains the exact same
content as f equals m a all I've done is

content as f equals m a all I've done is

content as f equals m a all I've done is
split it up into two pieces but working

split it up into two pieces but working

split it up into two pieces but working
with the first order equations has a

with the first order equations has a

with the first order equations has a
couple of big advantages to see why it's

couple of big advantages to see why it's

couple of big advantages to see why it's
helpful let's draw a picture with X on

helpful let's draw a picture with X on

helpful let's draw a picture with X on
the horizontal axis and P on the

the horizontal axis and P on the

the horizontal axis and P on the
vertical axis this diagram is called the

vertical axis this diagram is called the

vertical axis this diagram is called the
phase space and each point in this plane

phase space and each point in this plane

phase space and each point in this plane
tells us where the block is and what its

tells us where the block is and what its

tells us where the block is and what its
momentum is or equivalently its velocity

momentum is or equivalently its velocity

momentum is or equivalently its velocity
at any given moment so for example when

at any given moment so for example when

at any given moment so for example when
we pull the block out to its initial

we pull the block out to its initial

we pull the block out to its initial
position and then release it from rest

position and then release it from rest

position and then release it from rest
that initial State corresponds to this

that initial State corresponds to this

that initial State corresponds to this
point here on the horizontal axis where

point here on the horizontal axis where

point here on the horizontal axis where
x equals x Sub 0 and P is equal to zero

x equals x Sub 0 and P is equal to zero

x equals x Sub 0 and P is equal to zero
after we let it go the block is going to

after we let it go the block is going to

after we let it go the block is going to
begin to move and so these X and P

begin to move and so these X and P

begin to move and so these X and P
coordinates are going to change with

coordinates are going to change with

coordinates are going to change with
time so the point in this plane moves

time so the point in this plane moves

time so the point in this plane moves
around with time and it traces out a

around with time and it traces out a

around with time and it traces out a
curve called a float

curve called a float

curve called a float
and flow really is a good name for it

and flow really is a good name for it

and flow really is a good name for it
because I want you to Picture This Plane

because I want you to Picture This Plane

because I want you to Picture This Plane
like the surface of a pool of water with

like the surface of a pool of water with

like the surface of a pool of water with
some current flowing around it then we

some current flowing around it then we

some current flowing around it then we
take something like a ping pong ball say

take something like a ping pong ball say

take something like a ping pong ball say
and set it down at the point for our

and set it down at the point for our

and set it down at the point for our
initial conditions once we let it go the

initial conditions once we let it go the

initial conditions once we let it go the
current will carry the ball off moving

current will carry the ball off moving

current will carry the ball off moving
it around the surface of the water the

it around the surface of the water the

it around the surface of the water the
flow is the path that the ball follows

flow is the path that the ball follows

flow is the path that the ball follows
through the water but what determines

through the water but what determines

through the water but what determines
the shape and strength of the current

the shape and strength of the current

the shape and strength of the current
that's telling the ball where to move

that's telling the ball where to move

that's telling the ball where to move
our differential equations of course

our differential equations of course

our differential equations of course
it's helpful to write the pair of them

it's helpful to write the pair of them

it's helpful to write the pair of them
as a single vector equation again X and

as a single vector equation again X and

as a single vector equation again X and
P are the coordinates of the ping pong

P are the coordinates of the ping pong

P are the coordinates of the ping pong
ball on the surface of the water and so

ball on the surface of the water and so

ball on the surface of the water and so
their time derivative is telling us the

their time derivative is telling us the

their time derivative is telling us the
ball's velocity Vector at each point on

ball's velocity Vector at each point on

ball's velocity Vector at each point on
the surface over here at our initial

the surface over here at our initial

the surface over here at our initial
point x was positive and P was Zero then

point x was positive and P was Zero then

point x was positive and P was Zero then
the horizontal component of the Velocity

the horizontal component of the Velocity

the horizontal component of the Velocity
Vector is zero and the vertical

Vector is zero and the vertical

Vector is zero and the vertical
component is negative so the velocity of

component is negative so the velocity of

component is negative so the velocity of
the imaginary ping pong ball points

the imaginary ping pong ball points

the imaginary ping pong ball points
straight down at that point likewise we

straight down at that point likewise we

straight down at that point likewise we
can go to each point in this plane and

can go to each point in this plane and

can go to each point in this plane and
draw this velocity Vector those arrows

draw this velocity Vector those arrows

draw this velocity Vector those arrows
are what tell us the current that's

are what tell us the current that's

are what tell us the current that's
swirling around the plane and what

swirling around the plane and what

swirling around the plane and what
dictates how the ping pong ball will

dictates how the ping pong ball will

dictates how the ping pong ball will
move you can see that they're sort of

move you can see that they're sort of

move you can see that they're sort of
swirling around the origin here that's

swirling around the origin here that's

swirling around the origin here that's
the equilibrium point and I'm using the

the equilibrium point and I'm using the

the equilibrium point and I'm using the
colors here to indicate how strong the

colors here to indicate how strong the

colors here to indicate how strong the
current is its smallest for the yellow

current is its smallest for the yellow

current is its smallest for the yellow
arrows near the middle and gets bigger

arrows near the middle and gets bigger

arrows near the middle and gets bigger
for the Red Arrows farther out by

for the Red Arrows farther out by

for the Red Arrows farther out by
following those vectors starting from

following those vectors starting from

following those vectors starting from
our initial conditions we see that the

our initial conditions we see that the

our initial conditions we see that the
flow is an ellipse that wraps around the

flow is an ellipse that wraps around the

flow is an ellipse that wraps around the
origin again and again as the block

origin again and again as the block

origin again and again as the block
oscillates back and forth around

oscillates back and forth around

oscillates back and forth around
equilibrium this is definitely a more

equilibrium this is definitely a more

equilibrium this is definitely a more
abstract way of thinking about the

abstract way of thinking about the

abstract way of thinking about the
solution to our differential equation

solution to our differential equation

solution to our differential equation
remember the physical system here is the

remember the physical system here is the

remember the physical system here is the
block sliding back and forth on this

block sliding back and forth on this

block sliding back and forth on this
one-dimensional line so obviously there

one-dimensional line so obviously there

one-dimensional line so obviously there
isn't actually any pool of water or ping

isn't actually any pool of water or ping

isn't actually any pool of water or ping
pong ball those are just useful

pong ball those are just useful

pong ball those are just useful
mathematical constructs for picturing

mathematical constructs for picturing

mathematical constructs for picturing
what's going on but what this picture

what's going on but what this picture

what's going on but what this picture
buys us is that we can very quickly

buys us is that we can very quickly

buys us is that we can very quickly
understand what the motion of our system

understand what the motion of our system

understand what the motion of our system
is going to look like without solving

is going to look like without solving

is going to look like without solving
any differential equations all we need

any differential equations all we need

any differential equations all we need
to do is draw the arrows at each point

to do is draw the arrows at each point

to do is draw the arrows at each point
in face space that we get from the right

in face space that we get from the right

in face space that we get from the right
hand side of Hamilton's equations either

hand side of Hamilton's equations either

hand side of Hamilton's equations either
by hand or better yet on a computer

by hand or better yet on a computer

by hand or better yet on a computer
that's already an extremely useful way

that's already an extremely useful way

that's already an extremely useful way
of thinking about differential equations

of thinking about differential equations

of thinking about differential equations
but Hamilton's formulation also gives us

but Hamilton's formulation also gives us

but Hamilton's formulation also gives us
a really direct way of explicitly

a really direct way of explicitly

a really direct way of explicitly
writing down the solution at least for a

writing down the solution at least for a

writing down the solution at least for a
linear equation like the harmonic

linear equation like the harmonic

linear equation like the harmonic
oscillator and that's the last thing I

oscillator and that's the last thing I

oscillator and that's the last thing I
want to to quickly sketch out for you to

want to to quickly sketch out for you to

want to to quickly sketch out for you to
see how it works let's Express

see how it works let's Express

see how it works let's Express
Hamilton's equations as a matrix

Hamilton's equations as a matrix

Hamilton's equations as a matrix
equation this might look like we've made

equation this might look like we've made

equation this might look like we've made
things more complicated but hang on a

things more complicated but hang on a

things more complicated but hang on a
second we'll see how it pays off think

second we'll see how it pays off think

second we'll see how it pays off think
about a simple differential equation

about a simple differential equation

about a simple differential equation
like d by DT of Z equals Alpha times Z

like d by DT of Z equals Alpha times Z

like d by DT of Z equals Alpha times Z
for some constant Alpha this equation

for some constant Alpha this equation

for some constant Alpha this equation
says that when we take the derivative of

says that when we take the derivative of

says that when we take the derivative of
a function Z of T we're supposed to get

a function Z of T we're supposed to get

a function Z of T we're supposed to get
back Z again times a number Alpha and

back Z again times a number Alpha and

back Z again times a number Alpha and
the solution is simple it's just e to

the solution is simple it's just e to

the solution is simple it's just e to
the alpha T times Z of zero that's

the alpha T times Z of zero that's

the alpha T times Z of zero that's
because the derivative of the

because the derivative of the

because the derivative of the
exponential just turns back into itself

exponential just turns back into itself

exponential just turns back into itself
times a factor of Alpha from the chain

times a factor of Alpha from the chain

times a factor of Alpha from the chain
Rule and when you plug in t equals 0 you

Rule and when you plug in t equals 0 you

Rule and when you plug in t equals 0 you
get that initial value Z of zero but

get that initial value Z of zero but

get that initial value Z of zero but
notice that our Matrix equation for the

notice that our Matrix equation for the

notice that our Matrix equation for the
block on a spring is essentially of the

block on a spring is essentially of the

block on a spring is essentially of the
same form only with vectors and matrices

same form only with vectors and matrices

same form only with vectors and matrices
now instead of single numbers it says

now instead of single numbers it says

now instead of single numbers it says
that the derivative of the vector XP is

that the derivative of the vector XP is

that the derivative of the vector XP is
equal to itself multiplied by a constant

equal to itself multiplied by a constant

equal to itself multiplied by a constant
Matrix m

Matrix m

Matrix m
and the solution is just the Matrix

and the solution is just the Matrix

and the solution is just the Matrix
analog of our simple equation for Z we

analog of our simple equation for Z we

analog of our simple equation for Z we
take the initial Vector X of 0 P of 0

take the initial Vector X of 0 P of 0

take the initial Vector X of 0 P of 0
and act on it with the Matrix we get by

and act on it with the Matrix we get by

and act on it with the Matrix we get by
exponentiating T times m

exponentiating T times m

exponentiating T times m
that looks reasonable but of course we

that looks reasonable but of course we

that looks reasonable but of course we
have to ask ourselves what it even means

have to ask ourselves what it even means

have to ask ourselves what it even means
to take the exponential of a matrix here

to take the exponential of a matrix here

to take the exponential of a matrix here
and it's defined by the usual Taylor

and it's defined by the usual Taylor

and it's defined by the usual Taylor
series for E we get one plus the thing

series for E we get one plus the thing

series for E we get one plus the thing
in the exponent plus half the thing

in the exponent plus half the thing

in the exponent plus half the thing
squared plus one over three factorial

squared plus one over three factorial

squared plus one over three factorial
the thing cubed and so on that might

the thing cubed and so on that might

the thing cubed and so on that might
look like a nasty thing to try to

look like a nasty thing to try to

look like a nasty thing to try to
compute and it certainly can be in

compute and it certainly can be in

compute and it certainly can be in
general but for our Matrix m in this

general but for our Matrix m in this

general but for our Matrix m in this
problem the answer works out in a

problem the answer works out in a

problem the answer works out in a
beautiful and simple way again I'll show

beautiful and simple way again I'll show

beautiful and simple way again I'll show
you how to get it step by step in the

you how to get it step by step in the

you how to get it step by step in the
notes but here's the result

notes but here's the result

notes but here's the result
we get cosine of Omega T in the top left

we get cosine of Omega T in the top left

we get cosine of Omega T in the top left
and bottom right and sine of Omega T in

and bottom right and sine of Omega T in

and bottom right and sine of Omega T in
the top right and bottom left times some

the top right and bottom left times some

the top right and bottom left times some
constants

constants

constants
and finally when we plug in our initial

and finally when we plug in our initial

and finally when we plug in our initial
conditions here's what we get

conditions here's what we get

conditions here's what we get
lo and behold the first line tells us

lo and behold the first line tells us

lo and behold the first line tells us
that X of T is equal to x0 cosine Omega

that X of T is equal to x0 cosine Omega

that X of T is equal to x0 cosine Omega
T yet again and the second line is the

T yet again and the second line is the

T yet again and the second line is the
corresponding momentum M times the

corresponding momentum M times the

corresponding momentum M times the
derivative of x

derivative of x

derivative of x
so I hope I've convinced you of how

so I hope I've convinced you of how

so I hope I've convinced you of how
powerful Hamilton's method is of

powerful Hamilton's method is of

powerful Hamilton's method is of
converting a second order equation into

converting a second order equation into

converting a second order equation into
a pair of first order equations both for

a pair of first order equations both for

a pair of first order equations both for
explicitly solving the equation with the

explicitly solving the equation with the

explicitly solving the equation with the
Matrix exponential but also for

Matrix exponential but also for

Matrix exponential but also for
visualizing the behavior of the solution

visualizing the behavior of the solution

visualizing the behavior of the solution
as a flow on face space

as a flow on face space

as a flow on face space
all right I hope that was fun we got to

all right I hope that was fun we got to

all right I hope that was fun we got to
see how to solve the simple harmonic

see how to solve the simple harmonic

see how to solve the simple harmonic
oscillator equation with five different

oscillator equation with five different

oscillator equation with five different
increasingly sophisticated techniques

increasingly sophisticated techniques

increasingly sophisticated techniques
again nobody's saying you actually

again nobody's saying you actually

again nobody's saying you actually
should use Laplace transforms or Matrix

should use Laplace transforms or Matrix

should use Laplace transforms or Matrix
exponentials to solve such a simple

exponentials to solve such a simple

exponentials to solve such a simple
differential equation but as you work

differential equation but as you work

differential equation but as you work
your way up in physics you're quickly

your way up in physics you're quickly

your way up in physics you're quickly
going to start running into more

going to start running into more

going to start running into more
challenging differential equations where

challenging differential equations where

challenging differential equations where
the methods you've gotten a glimpse of

the methods you've gotten a glimpse of

the methods you've gotten a glimpse of
here become invaluable

here become invaluable

here become invaluable
remember that you can get the notes for

remember that you can get the notes for

remember that you can get the notes for
this video for free at the link in the

this video for free at the link in the

this video for free at the link in the
description and I'll also put the links

description and I'll also put the links

description and I'll also put the links
to all those other videos that I've

to all those other videos that I've

to all those other videos that I've
mentioned down there I want to say a

mentioned down there I want to say a

mentioned down there I want to say a
huge thank you to my supporters on

huge thank you to my supporters on

huge thank you to my supporters on
patreon if you want to see more videos

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like this you can join too at the link

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up in the corner also this was the first

up in the corner also this was the first

up in the corner also this was the first
video I've made in large part using

video I've made in large part using

video I've made in large part using
manim the programming library for math

manim the programming library for math

manim the programming library for math
animations created by three blue one

animations created by three blue one

animations created by three blue one
brown and further developed by the

brown and further developed by the

brown and further developed by the
brilliant people who work on the open

brilliant people who work on the open

brilliant people who work on the open
source project I want to say another

source project I want to say another

source project I want to say another
huge thank you to them for sharing all

huge thank you to them for sharing all

huge thank you to them for sharing all
their hard work thank you so much for

their hard work thank you so much for

their hard work thank you so much for
watching and I'll see you soon with

watching and I'll see you soon with

watching and I'll see you soon with
another physics lesson