Cookies   I display ads to cover the expenses. See the privacy policy for more information. You can keep or reject the ads.

Video thumbnail
Quoting Steven Strogatz, “Since Newton, mankind has come to realize that the laws
of physics are always expressed in the language of differential equations.” Of course, this
language is spoken well beyond the boundaries of physics as well, and being able to speak
it and read it adds a new color to how you view the world around you.
In the next few videos, I want to give a sort of tour of this topic. To aim is to give a
big picture view of what this part of math is all about, while at the same time being
happy to dig into the details of specific examples as they come along.
I’ll be assuming you know the basics of calculus, like what derivatives and integrals
are, and in later videos we’ll need some basic linear algebra, but not much beyond
that.
Differential equations arise whenever it’s easier to describe change than absolute amounts.
It’s easier to say why population sizes grow or shrink than it is to describe why
the have the particular values they do at some point in time; It may be easier to describe
why your love for someone is changing than why it happens to be where it is now. In physics,
more specifically Newtonian mechanics, motion is often described in terms of force. Force
determines acceleration, which is a statement about change.
These equations come in two flavors; Ordinary differential equations, or ODEs, involving
functions with a single input, often thought of as time, and Partial differential equations,
or PDEs, dealing with functions that have multiple inputs. Partial derivatives are something
we’ll look at more closely in the next video; you often think of them involving a whole
continuum of values changing with time, like the temperature of every point in a solid
body, or the velocity of a fluid at every point in space. Ordinary differential equations,
our focus for now, involve only a finite collection of values changing with time.
It doesn’t have to be time, per se, your one independent variable could be something
else, but things changing with time are the prototypical and most common examples of differential
equations. Physics (simple)
Physics offers a nice playground for us here, with simple examples to start with, and no
shortage of intricacy and nuance as we delve deeper.
As a nice warmup, consider the trajectory of something you throw in the air. The force
of gravity near the surface of the earth causes things to accelerate downward at 9.8 m/s per
second. Now unpack what that really means: If you look at some object free from other
forces, and record its velocity every second, these vectors will accrue an additional downward
component of 9.8 m/s every second. We call this constant 9.8 “g”.
This gives an example of a differential equation, albeit a relatively simple one. Focus on the
y-coordinate, as a function of time. It’s derivative gives the vertical component of
velocity, whose derivative in turn gives the vertical component of acceleration. For compactness,
let’s write this first derivative as y-dot, and the second derivative as y-double-dot.
Our equation is simply y-double-dot = -g. This is one where you can solve by integrating,
which is essentially working backwards. First, what is velocity, what function has -g as
a derivative? Well, -g*t. Or rather, -g*t + (the initial velocity). Notice that you
have this degree of freedom which is determined by an initial condition. Now what function
has this as a derivative? -(½)g*t^2 + v_0 * t. Or, rather, add in a constant based on
whatever the initial position is.
Things get more interesting when the forces acting on a body depend on where that body
is. For example, studying the motion of planets, stars and moons, gravity can no longer be
considered a constant. Given two bodies, the pull on one is in the direction of the other,
with a strength inversely proportional to the square of the distance between them.
As always, the rate of change of position is velocity, but now the rate of change of
velocity is some function of position. The dance between these mutually-interacting variables
is mirrored in the dance between the mutually-interacting bodies which they describe.
So often in differential equations, the puzzles you face involve finding a function whose
derivative and/or higher order derivatives are defined in terms of itself.
In physics, it’s most common to work with second order differential equations, which
means the highest derivative you find in the expression here is a second derivative. Higher
order differential equations would be ones with third derivatives, fourth derivatives
and so on; puzzles with more intricate clues.
The sensation here is one of solving an infinite continuous jigsaw puzzle. In a sense you have
to find infinitely many numbers, one for each point in time, constrained by a very specific
way that these values intertwine with their own rate of change, and the rate of change
of that rate of change.
I want you to take some time digging in to a deceptively simple example: A pendulum.
How does this angle theta that it makes with the vertical change as a function of time.
This is often given as an example in introductory physics classes of harmonic motion, meaning
it oscillates like a sine wave. More specifically, one with a period of 2pi * L/g, where L is
the length of the pendulum, and g is gravity.
However, these formulas are actually lies. Or, rather, approximations which only work
in the realm of small angles. If you measured an actual pendulum, you’d find that when
you pull it out farther, the period is longer than what that high-school physics formulas
would suggest. And when you pull it really far out, the value of theta vs. time doesn’t
even look like a sine wave anymore.
First thing’s first, let’s set up the differential equation. We’ll measure its
position as a distance x along this arc. If the angle theta we care about is measured
in radians, we can write x and L*theta, where L is the length of the pendulum.
As usual, gravity pulls down with acceleration g, but because the pendulum constrains the
motion of this mass, we have to look at the component of this acceleration in the direction
of motion. A little geometry exercise for you is to show that this little angle here
is the same as our theta. So the component of gravity in the direction of motion, opposite
this angle, will be -g*sin(theta).
Here we’re considering theta to be positive when the pendulum is swung to the right, and
negative when it’s swung to the left, and this negative sign in the acceleration indicates
that it’s always pointed in the opposite direction from displacement.
So the second derivative
of x, the acceleration, is -g*sin(theta). Since x is L*theta, that means the second
derivative of theta is -(g/L) * sin(theta). To be somewhat more realistic, let’s add
in a term to account for air resistance, which perhaps we model as being proportional to
the velocity. We write this as -mu * theta-dot, where -mu is some constant determining how
quickly the pendulum loses energy.
This is a particularly juicy differential equation. Not easy to solve, but not so hard
that we can’t reasonably get some meaningful understanding of it.
At first you might think that this sine function relates to the sine wave pattern for the pendulum.
Ironically, though, what you'll eventually find is that the opposite is true. The presence
of the sine in this equation is precisely why the real pendulum doesn't oscillate with
the sine wave pattern.
If that sounds odd, consider the fact that here, the sine function takes theta as an
input, but the approximate solution has the value theta itself oscillating as a sine wave.
Clearly something fishy is afoot.
One thing I like about this example is that even though it’s comparatively simple, it
exposes an important truth about differential equations that you need to be grapple with:
They’re really freaking hard to solve.
In this case, if we remove the damping term, we can just barely write down an analytic
solution, but it’s hilariously complicated, involving all these functions you’re probably
never heard of written in terms of integrals and weird inverse integral problems.
Presumably, the reason for finding a solution is to then be able to make computations, and
to build an understanding for whatever dynamics your studying. In a case like this, those
questions have just been punted off to figuring out how to compute and understand these new
functions.
And more often, like if we add back this dampening term, there is not a known way to write down
an exact solution analytically. Well, for any hard problem you could just define a new
function to be the answer to that problem. Heck, even name it after yourself if you want.
But again, that’s pointless unless it leads you to being able to compute and understand
the answer.
So instead, in studying differential equations, we often do a sort of short-circuit and skip
the actual solution part, and go straight to building understanding and making computations
from the equations alone. Let me walk through what that might look like with the Pendulum.
Phase space What do you hold in your head, or what visualization
could you get some software to pull up for you, to understand the many possible ways
a pendulum governed by these laws might evolve depending on its starting conditions?
You might be tempted to try imagining the graph of theta(t), and somehow interpreting
how its position, slope, and curvature all inter-relate. However, what will turn out
to be both easier and more general is to start by visualizing all possible states of the
system in a 2d plane.
The state of the pendulum can be fully described by two numbers, the angle, and the angular
velocity. You can freely change these two values without necessarily changing the other,
but the acceleration is purely a function of these two values. So each point of this
2d plane fully describes the pendulum at a given moment. You might think of these as
all possible initial conditions of the pendulum. If you know this initial angle and angular
velocity, that’s enough to predict how the system will evolve as time moves forward.
If you haven’t worked with them, these sorts of diagrams can take a little getting used
to. What you’re looking at now, this inward spiral, is a fairly typical trajectory for
our pendulum, so take a moment to think carefully about what’s being represented. Notice how
at the start, as theta decreases, theta-dot gets more negative, which makes sense because
the pendulum moves faster in the leftward direction as it approaches the bottom. Keep
in mind, even though the velocity vector on this pendulum is pointed to the left, the
value of that velocity is being represented by the vertical component of our space. It’s
important to remind yourself that this state space is abstract, and distinct from the physical
space where the pendulum lives and moves.
Since we’re modeling it as losing some energy to air resistance, this trajectory spirals
inward, meaning the peak velocity and displacement each go down by a bit with each swing. Our
point is, in a sense, attracted to the origin where theta and theta-dot both equal 0.
With this space, we can visualize a differential equation as a vector field. Here, let me show
you what I mean.
The pendulum state is this vector, [theta, theta-dot]. Maybe you think of it as an arrow,
maybe as a point; what matters is that it has two coordinates, each a function of time.
Taking the derivative of that vector gives you its rate of change; the direction and
speed that it will tend to move in this diagram. That derivative is a new vector, [theta-dot,
theta-double-dot], which we visualize as being attached to the relevant point in this space.
Take a moment to interpret what this is saying.
The first component for this rate-of-change vector is theta-dot, so the higher up we are
on the digram, the more the point tends to move to the right, and the lower we are, the
more it tends to move to the left. The vertical component is theta-double-dot, which our differential
equation lets us rewrite entirely in terms of theta and theta-dot. In other words, the
first derivative of our state vector is some function of that vector itself.
Doing the same at all points of this space will show how the state tends to change from
any position, artificially scaling down the vectors when we draw them to prevent clutter,
but using color to loosely indicate magnitude.
Notice that we’ve effectively broken up a single second order equation into a system
of two first order equations. You might even give theta-dot a different name to emphasize
that we’re thinking of two separate values, intertwined via this mutual effect they have
on one and other’s rate of change. This is a common trick in the study of differential
equations, instead of thinking about higher order changes of a single value, we often
prefer to think of the first derivative of vector values.
In this form, we have a nice visual way to think about what solving our equation means:
As our system evolves from some initial state, our point in this space will move along some
trajectory in such a way that at every moment, the velocity of that point matches the vector
from this vector field. Keep in mind, this velocity is not the same thing as the physical
velocity of our pendulum. It’s a more abstract rate of change encoding the changes in both
theta and theta-dot.
You might find it fun to pause for a moment and think through what exactly some of these
trajectory lines say about possible ways the pendulum evolves for different starting conditions.
For example, in regions where theta-dot is quite high, the vectors guide the point to
travel to the right quite a ways before settling down into an inward spiral. This corresponds
to a pendulum with a high initial velocity, fully rotating around several times before
settling down into a decaying back and forth.
Having a little more fun, when I tweak this air resistance term mu, say increasing it,
you can immediately see how this will result in trajectories that spiral inward faster,
which is to say the pendulum slows down faster. Imagine you saw the equations out of context,
not knowing they described a pendulum; it’s not obvious just-looking at them that increasing
the value of mu means the system tends towards some attracting state faster, so getting some
software to draw these vector fields for you can be a great way to gain an intuition for
how they behave.
What’s wonderful is that any system of ordinary differential equations can be described by
a vector field like this, so it’s a very general way to get a feel for them.
Usually, though, they have many more dimensions. For example, consider the famous three-body
problem, which is to predict how three masses in 3d space will evolve if they act on each
other with gravity, and you know their initial positions and velocities.
Each mass has three coordinates describing its position and three more describing its
momentum, so the system has 18 degrees of freedom, and hence an 18-dimensional space
of possible states. It’s a bizarre thought, isn’t it? A single point meandering through
and 18-dimensional space we cannot visualize, obediently taking steps through time based
on whatever vector it happens to be sitting on from moment to moment, completely encoding
the positions and momenta of 3 masses in ordinary, physical, 3d space.
(In practice, by the way, you can reduce this number of dimension by taking advantage of
the symmetries in your setup, but the point of more degrees of freedom resulting in a
higher-dimensional state space remains the same).
In math, we often call a space like this a “phase space”. You’ll hear me use the
term broadly for spaces encoding all kinds of states for changing systems, but you should
know that in the context of physics, especially Hamiltonian mechanics, the term is often reserved
for a special case. Namely, a space whose axes represent position and momentum.
So a physicist would agree that the 18-dimension space describing the 3-body problem is a phase
space, but they might ask that we make a couple of modifications to our pendulum set up for
it to properly deserve the term. For those of you who watched the block collision videos,
the planes we worked with there would happily be called phase spaces by math folk, though
a physicist might prefer other terminology. Just know that the specific meaning may depend
on your context.
It may seem like a simple idea, depending on how well indoctrinated you are to modern
ways of thinking about math, but it’s worth keeping in mind that it took humanity quite
a while to really embrace thinking of dynamics spatially like this, especially when the dimensions
get very large. In his book Chaos, James Gleick describes phase space as “one of the most
powerful inventions of modern science.”
One reason it’s powerful is that you can ask questions not just about a single initial
state, but a whole spectrum of initial states. The collection of all possible trajectories
is reminiscent of a moving fluid, so we call it phase flow.
To take one example of why phase flow is a fruitful formulation, the origin of our space
corresponds to the pendulum standing still; and so does this point over here, representing
when the pendulum is balanced upright. These are called fixed points of the system, and
one natural question to ask is whether they are stable. That is, will tiny nudges to the
system result in a state that tends back towards the stable point or away from it. Physical
intuition for the pendulum makes the answer here obvious, but how would you think about
stability just by looking at the equations, say if they arose from some completely different
and less intuitive context?
We’ll go over how to compute the answer to a question like this in following videos,
and the intuition for the relevant computations are guided heavily by the thought of looking
at a small region in this space around the fixed point and asking about whether the flow
contracts or expands its points.
Speaking of attraction and stability, let’s take a brief sidestep to talk about love.
The Strogatz quote I referenced earlier comes from a whimsical column in the New York Times
on mathematical models of love, an example well worth pilfering to illustrate that we’re
not just talking about physics.
Imagine you’ve been flirting with someone, but there’s been some frustrating inconsistency
to how mutual the affections seem. And perhaps during a moment when you turn your attention
towards physics to keep your mind off this romantic turmoil, mulling over your broken
up pendulum equations, you suddenly understand the on-again-off-again dynamics of your flirtation.
You’ve noticed that your own affections tend to increase when your companion seems
interested in you, but decrease when they seem colder. That is, the rate of change for
your love is proportional to their feelings for you.
But this sweetheart of yours is precisely the opposite: Strangely attracted to you when
you seem uninterested, but turned off once you seem too keen.
The phase space for these equations looks very similar to the center part of your pendulum
diagram. The two of you will go back and forth between affection and repulsion in an endless
cycle. A metaphor of pendulum swings in your feelings would not just be apt, but mathematically
verified. In fact, if your partner’s feelings were further slowed when they feel themselves
too in love, let’s say out of a fear of being made vulnerable, we’d have a term
matching the friction of your pendulum, and you two would be destined to an inward spiral
towards mutual ambivalence. I hear wedding bells already.
The point is that two very different-seeming laws of dynamics, one from physics initially
involving a single variable, and another from...er...chemistry with two variables, actually have a very similar
structure, easier to recognize when looking at their phase spaces. Most notably, even
though the equations are different, for example there’s no sine in your companion’s equation,
the phase space exposes an underlying similarity nevertheless.
In other words, you’re not just studying a pendulum right now, the tactics you develop
to study one case have a tendency to transfer to many others.
Okay, so phase diagrams are a nice way to build understanding, but what about actually
computing the answer to our equation? Well, one way to do this is to essentially simulate
what the world will do, but using finite time steps instead of the infinitesimals and limits
defining calculus.
The basic idea is that if you’re at some point on this phase diagram, take a step based
on whatever vector your sitting on for some small time step, delta-t. Specifically, take
a step of delta-T times that vector. Remember, in drawing this vector field, the magnitude
of each vector has been artificially scaled down to prevent clutter. Do this repeatedly,
and your final location will be an approximation of theta(t), where t is the sum of all your
time steps.
If you think about what’s being shown right now, and what that would imply for the pendulum’s
movement, you’d probably agree it’s grossly inaccurate. But that’s just because the
timestep delta-t of 0.5 is way too big. If we turn it down, say to 0.01, you can get
a much more accurate approximation, it just takes many more repeated steps is all. In
this case, computing theta(10) requires a thousand little steps. Luckily, we live in
a world with computers, so repeating a simple task 1,000 times is as simple as articulating
that task with a programming language.
In fact, let’s write a little python program that computes theta(t) for us. It will make
use of the differential equation, which returns the second derivative of theta as a function
of theta and theta-dot. You start by defining two variables, theta and theta-dot, in terms
of some initial values. In this case I’ll choose pi / 3, which is 60-degrees, and 0
for the angular velocity.
Next, write a loop which corresponds to many little time steps between 0 and 10, each of
size delta-t, which I’m setting to be 0.01 here. In each step of the loop, increase theta
by theta-dot times delta-t, and increase theta-dot by theta-double-dot times delta-t, where theta-double-dot
can be computed based on the differential equation. After all these little steps, simple
return the value of theta.
This is called solving the differential equation numerically. Numerical methods can get way
more sophisticated and intricate to better balance the tradeoff between accuracy and
efficiency, but this loop gives the basic idea.
So even though it sucks that we can’t always find exact solutions, there are still meaningful
ways to study differential equations in the face of this inability.
In the following videos, we will look at several methods for finding exact solutions when it’s
possible. But one theme I’d like to focus is on is how these exact solutions can also
help us study the more general unsolvable cases.
But it gets worse. Just as there is a limit to how far exact analytic solutions can get
us, one of the great fields to have emerged in the last century, chaos theory, has exposed
that there are further limits on how well we can use these systems for prediction, with
or without exact solutions. Specifically, we know that for some systems, small variations
to the initial conditions, say the kind due to necessarily imperfect measurements, result
in wildly different trajectories. We’ve even built some good understanding for why
this happens. The three body problem, for example, is known to have seeds of chaos within
it.
So looking back at that quote from earlier, it seems almost cruel of the universe to fill
its language with riddles that we either can’t solve, or where we know that any solution
would be useless for long-term prediction anyway. It is cruel, but then again, that
should be reassuring. It gives some hope that the complexity we see in the world can be
studied somewhere in the math, and that it’s not hidden away in some mismatch between model
and reality.