Differential equations, a tourist's guide | DE1

Summary

Este video profundiza en el estudio de ecuaciones diferenciales, presentadas como la lengua universal de las leyes de la física. A través de ejemplos como el movimiento de un pendulo y la trayectoria de un objeto lanzado, se explica cómo estas ecuaciones describen el cambio en lugar de valores absolutos. Se introducen ecuaciones diferenciales ordinarias y parciales, y se discute cómo尽管 no siempre es posible encontrar soluciones exactas, existen métodos numéricos efectivos para aproximarse a las soluciones. Además, se toca el tema de la caótica dinámica en sistemas complejos, como el problema de tres cuerpos.

Takeaways

• 📚 Las ecuaciones diferenciales son una herramienta fundamental en la descripción de cambios y tienen aplicaciones más allá de la física.
• 🌐 Las ecuaciones diferenciales se dividen en ecuaciones diferenciales ordinarias (ODEs) y ecuaciones diferenciales parciales (PDEs).
• 📈 Las ODEs involucran funciones de una variable, a menudo el tiempo, mientras que las PDEs manejan múltiples entradas.
• 🚀 Las ecuaciones diferenciales surgen cuando es más fácil describir el cambio que las cantidades absolutas.
• 🔄 Las soluciones de las ecuaciones diferenciales pueden requerir técnicas de integración y análisis de vectores.
• 📊 La visualización de las soluciones a través de diagramas de fase o espacios de estados puede proporcionar una comprensión más intuitiva del comportamiento del sistema.
• 🌐 En el espacio de estados, cada punto describe un estado posible del sistema, y el movimiento a lo largo del tiempo se guía por vectores que representan el cambio.
• 🔢 A pesar de que no siempre es posible encontrar soluciones analíticas exactas, los métodos numéricos ofrecen una forma efectiva de aproximarse a las soluciones.
• 💻 Los programas de computadora son útiles para simular la evolución de sistemas dinámicos a través del tiempo utilizando métodos numéricos.
• 🌀 El concepto de caos teórico muestra que hay límites en la capacidad de predicción a largo plazo, incluso con soluciones exactas.
• 💡 A pesar de los desafíos, el estudio de las ecuaciones diferenciales y la teoría del caos ofrecen una ventana al entendimiento de la complejidad en el mundo natural.

Q & A

¿Qué es una ecuación diferencial y cómo surgen en la física?

Una ecuación diferencial es una方程 que describe la relación entre una función y sus derivadas. En la física, surgen cuando es más fácil describir el cambio que las cantidades absolutas, como la variación de la población o la aceleración de un objeto debido a la fuerza.

¿Cuáles son las dos variedades de ecuaciones diferenciales mencionadas en el script?

Las dos variedades de ecuaciones diferenciales son las ecuaciones diferenciales ordinarias (ODEs) y las ecuaciones diferenciales parciales (PDEs). Las ODEs involucran funciones con una sola entrada, a menudo consideradas como el tiempo, mientras que las PDEs lidian con funciones que tienen múltiples entradas.

¿Qué es la aceleración debido a la gravedad y cómo se relaciona con la ecuación diferencial simple?

La aceleración debido a la gravedad es la componente vertical de la aceleración causada por la fuerza de gravedad cerca de la superficie de la Tierra, que es de 9.8 metros por segundo cuadrado. Esta aceleración se relaciona con una ecuación diferencial simple donde la segunda derivada de la función de posición (y-doble-punto) es igual a la constante negativa g.

¿Cómo se puede resolver una ecuación diferencial y cuál es el significado de integrar?

Se puede resolver una ecuación diferencial integrándola, que es esencialmente trabajar la pregunta en sentido inverso. En el contexto del script, se resuelve una ecuación diferencial encontrando una función que tenga una derivada dada, como la aceleración debido a la gravedad, y luego añadiendo condiciones iniciales para determinar la función completamente.

¿Qué es un vector campo y cómo se relaciona con las ecuaciones diferenciales?

Un vector campo es una representación visual de cómo un sistema evoluciona a lo largo del tiempo, donde cada punto en un espacio de estados tiene asociado un vector que describe su tasa de cambio. En el contexto de las ecuaciones diferenciales, un vector campo puede representar la evolución de un sistema en un diagrama de fase, mostrando cómo se mueve un punto que representa el estado del sistema a lo largo del tiempo según las leyes dada por la ecuación diferencial.

¿Qué es el concepto de fase y por qué es importante en la física y la matemática?

El concepto de fase se refiere al espacio que codifica todos los estados posibles de un sistema en cambio, con ejes que representan diferentes variables que describen el estado. Es importante en la física y la matemática porque permite analizar y visualizar el comportamiento de sistemas dinámicos en una representación abstracta que puede revelar patrones y propiedades que no son tan evidentes a través de las ecuaciones alones.

¿Cómo se puede aproximar la solución de una ecuación diferencial cuando no se puede encontrar una solución analítica?

Cuando no se puede encontrar una solución analítica para una ecuación diferencial, se puede utilizar métodos numéricos para aproximar la solución. Esto implica tomar pequeños pasos de tiempo y calcular la posición y velocidad en esos pasos utilizando la ecuación diferencial, lo que da como resultado una aproximación de la trayectoria del sistema en el espacio de estados.

¿Qué es la aproximación de Euler y cómo se utiliza para resolver ecuaciones diferenciales?

La aproximación de Euler es un método numérico simple para resolver ecuaciones diferenciales. Básicamente, consiste en tomar un paso de tiempo pequeño (delta t) y actualizar la posición y velocidad del sistema en función de las ecuaciones de la derivada y la segunda derivada, multiplicadas por delta t. Esto se repite para cada paso de tiempo, dando una aproximación de la trayectoria del sistema.

¿Qué es el caos teórico y cómo afecta la predictibilidad en sistemas dinámicos?

El caos teórico es un campo de estudio que se centra en sistemas dinámicos que son sensibles a las condiciones iniciales. Significa que pequeñas variaciones en las condiciones iniciales pueden llevar a resultados muy diferentes en el largo plazo, lo que hace que la predicción a largo plazo sea impracticable. Este concepto ha demostrado que hay límites fundamentales en nuestra capacidad para predecir el comportamiento de ciertos sistemas, incluso con una comprensión perfecta de las leyes que gobiernan su comportamiento.

¿Cómo se puede modelar la resistencia al movimiento de un péndulo mediante una ecuación diferencial?

La resistencia al movimiento de un péndulo se puede modelar añadiendo un término de resistencia al aire al ecuación diferencial que describe su movimiento. Este término es proporcional a la velocidad angular del péndulo y se representa como una fuerza negativa multiplicada por una constante (mu), que encapsula la resistencia al aire y la fricción.

¿Qué es un punto fijo en un sistema dinámico y cómo se relaciona con la estabilidad?

Un punto fijo en un sistema dinámico es un estado en el que el sistema no cambia con el tiempo. En relación con la estabilidad, los puntos fijos pueden ser estables o inestables. Un punto fijo es estable si las pequeñas perturbaciones al sistema resultan en estados que tienden a regresar al punto fijo; si son inestables, las perturbaciones hacen que el sistema se aleje del punto fijo.

¿Cómo se puede usar un diagrama de fase para entender el comportamiento de un sistema dinámico?

Un diagrama de fase se puede usar para visualizar cómo un sistema dinámico evoluciona a lo largo del tiempo. Cada punto en el diagrama representa un estado del sistema en un momento dado, y las trayectorias en el diagrama muestran cómo el estado del sistema cambia con el tiempo. Esto permite identificar patrones en el comportamiento del sistema, como ciclos repetitivos o tendencias a un estado de equilibrio.

¿Qué es la resolución numérica de ecuaciones diferenciales y cómo se realiza?

La resolución numérica de ecuaciones diferenciales es un método para encontrar aproximaciones a las soluciones de una ecuación diferencial cuando no se puede encontrar una solución exacta analítica. Se realiza mediante la aproximación de Euler o métodos similares, donde se toman pequeños pasos de tiempo y se actualiza la posición y velocidad del sistema en función de la ecuación diferencial en cada paso.

Outlines

00:00

📚 Introducción a las Ecuaciones Diferenciales

Este párrafo introduce el concepto de ecuaciones diferenciales, destacando su importancia y aplicación en la comprensión del mundo a nuestro alrededor. Stephen Strogatz menciona que las ecuaciones diferenciales son la forma en que la naturaleza expresa sus leyes fundamentales. El vídeo tiene como objetivo brindar una visión general de este tema matemático, explorando tanto conceptos básicos como ejemplos específicos. Se espera que el espectador tenga un conocimiento previo de cálculo y algebra lineal. Además, se definen los tipos de ecuaciones diferenciales: ecuaciones diferenciales ordinarias (ODEs) y ecuaciones diferenciales parciales (PDEs), con una breve descripción de cada una.

05:02

🔢 Orden y Derivadas en las Ecuaciones Diferenciales

En este párrafo se aborda la distinción entre ecuaciones diferenciales de segundo orden y las de ordenes más altos. Las ecuaciones de segundo orden son comunes en la física y involucran la maxima derivada de una función. Se describe la sensación de resolver estas ecuaciones como intentar ensamblar un rompecabezas continuo infinito, donde se busca una función cuya derivada está definida en términos de la función misma. Se utiliza el ejemplo simple de un péndulo para ilustrar cómo la posición y la velocidad están intrínsecamente relacionadas, y se establece la base para la discusión sobre ecuaciones diferenciales no resueltas en el vídeo continuidad.

10:05

🌐 Visualización de la Dinámica de un Péndulo

Este párrafo se centra en la visualización de la dinámica de un péndulo mediante el uso de un diagrama de fase, que es un espacio en dos dimensiones donde cada punto representa un estado posible del péndulo. Se describe cómo las ecuaciones diferenciales pueden ser visualizadas como un campo vectorial, donde cada punto en el diagrama de fase tiene un vector asociado que describe la tasa de cambio del sistema en esa posición. A través de esta visualización, se puede comprender cómo un sistema evoluciona a partir de un estado inicial, y se puede explorar cómo las trayectorias varían con diferentes condiciones iniciales.

15:07

💡 Comprensión de las Soluciones y la Fase del Sistema

Este párrafo discute la idea de visualizar las soluciones de un sistema dinámico como un flujo en el espacio de fase, y cómo las soluciones fijas (puntos fijos) pueden ser evaluadas en términos de estabilidad. Se introduce la noción de flujo de fase y se describe cómo este puede ser utilizado para investigar conceptos como la estabilidad de un sistema. Además, se hace una analogía entre la dinámica de un péndulo y la dinámica de las relaciones interpersonales, mostrando cómo las mismas ideas matemáticas pueden aplicarse a contextos muy distintos.

20:10

🔢 Métodos Numéricos para Resolver Ecuaciones Diferenciales

Este párrafo explora el método numérico para resolver ecuaciones diferenciales cuando no se pueden encontrar soluciones exactas analíticamente. Se describe el proceso de avanzar en pequeños pasos de tiempo, donde se actualiza la posición y velocidad del sistema en cada paso utilizando las ecuaciones diferenciales. Se menciona que, aunque este método puede ser menos preciso que una solución analítica, aún proporciona una forma significativa de estudiar y comprender el comportamiento de un sistema. Se presenta un ejemplo de cómo programar este método en Python para calcular la posición de un péndulo en un momento dado.

25:12

🌀 La Limitación de las Soluciones y el surgimiento de la Teoría del Caos

Este párrafo discute las limitaciones inherentes a las soluciones exactas de ecuaciones diferenciales y cómo la teoría del caos ha demostrado que incluso con soluciones, las predicciones a largo plazo pueden ser impracticables debido a la sensibilidad a las condiciones iniciales. Se menciona el problema de los tres cuerpos como un ejemplo de un sistema caótico. A pesar de estas limitaciones, se destaca la importancia de estudiar estas sistemas, ya que ofrecen una ventana para entender la complejidad del mundo natural.

Keywords

💡Ecuaciones diferenciales

Las ecuaciones diferenciales son expresiones matemáticas que describen cómo una cantidad cambia con el tiempo o con otra variable. En el video, se mencionan como una herramienta fundamental para entender los cambios en sistemas físicos y otros campos, y se explica cómo se relacionan con el lenguaje de las matemáticas aplicadas a la realidad.

💡Integración

El proceso de integración es el inverso del cálculo de derivadas, y se utiliza para encontrar una función que, al derivarse, proporcione una ecuación dada. En el contexto del video, la integración se utiliza para resolver ecuaciones diferenciales, encontrando la función que describe el cambio de una cantidad con el tiempo.

💡Derivadas

Las derivadas son una medida matemática de cómo una función cambia a medida que su argumento varía. En el video, las derivadas son importantes para entender y modelar el cambio en sistemas dinámicos, como el movimiento de un objeto bajo la influencia de la gravedad.

💡Mecánica newtoniana

La mecánica newtoniana es un conjunto de leyes que describen el movimiento de objetos debido a fuerzas, como la gravedad. En el video, se utiliza como un ejemplo práctico de cómo las ecuaciones diferenciales pueden aplicarse a fenómenos físicos y predecir el comportamiento de los cuerpos en movimiento.

💡Órbita de un pendulo

La órbita de un pendulo es el camino que sigue un objeto colgado de un pendulo mientras oscila. En el video, se utiliza para ilustrar cómo las ecuaciones diferenciales pueden ser utilizadas para predecir y entender el comportamiento de un sistema oscilante, aunque también se menciona que las aproximaciones simples no capturan todo el comportamiento real del pendulo.

💡Resistencia al movimiento

La resistencia al movimiento es una fuerza que opone el cambio de posición de un objeto. En el video, se menciona como un factor que influye en el movimiento del pendulo, y se utiliza para建模更加现实的行为 del pendulo, incluyendo la pérdida de energía debido a la fricción y el aire.

💡Sistemas dinámicos

Los sistemas dinámicos son sistemas que cambian con el tiempo de acuerdo con ciertas reglas. En el video, las ecuaciones diferenciales se utilizan para modelar y entender estos sistemas, permitiendo predecir su comportamiento a futuro a partir de condiciones iniciales.

💡Espacio de estados

El espacio de estados es un modelo abstracto que representa todos los posibles estados de un sistema dinámico en una serie de dimensiones. En el video, se utiliza para visualizar y entender cómo un sistema evoluciona a lo largo del tiempo, y cómo las condiciones iniciales afectan su trayectoria.

💡Cálculo numérico

El cálculo numérico es un método de aproximación para resolver problemas matemáticos que no se pueden solucionar exactamente. En el video, se utiliza para resolver ecuaciones diferenciales, proporcionando una aproximación del comportamiento del sistema modelado a través de pequeños pasos de tiempo.

💡Caos

El caos se refiere a un comportamiento de sistemas dinámicos en los que pequeñas variaciones en las condiciones iniciales pueden llevar a resultados altamente impredecibles. En el video, se menciona como un límite fundamental en la capacidad de predicción de sistemas, incluso con conocimiento de las ecuaciones que los gobiernan.

Highlights

Differential equations are a fundamental language of physics, expressing the laws of nature.

The language of differential equations extends beyond physics, influencing how we view the world.

Differential equations are useful for describing change rather than absolute amounts, such as population sizes or emotions.

In physics, force determines acceleration, which is a statement about change and is described by differential equations.

There are two types of differential equations: ordinary differential equations (ODEs) and partial differential equations (PDEs).

ODEs involve functions with a single input, often time, while PDEs deal with functions that have multiple inputs.

The trajectory of an object thrown in the air can be described using a simple differential equation involving gravity.

Differential equations can be solved by integrating, which works the problem backwards to find the original function.

The motion of celestial bodies like planets and stars involves complex differential equations where the forces depend on the bodies' positions.

Second-order differential equations are common in physics, representing the highest derivative in the expression.

Solving differential equations can feel like solving an infinite continuous jigsaw puzzle, with many numbers to find and intricate relationships.

The pendulum example shows that simple harmonic motion approximations only work for small angles and can be more complex in reality.

The presence of the sine function in the pendulum's differential equation is what causes real pendulums not to oscillate with a sine wave pattern.

Differential equations can be challenging to solve, and sometimes we focus on building understanding and making computations without finding an exact solution.

Vector fields and phase spaces provide a visual way to understand the dynamics of differential equations and their trajectories.

Phase flow and fixed points in a system can help us analyze stability and the long-term behavior of a system.

Numerical methods allow us to approximate solutions to differential equations by simulating the system's evolution in small time steps.

Chaos theory reveals that some systems are sensitive to initial conditions, leading to unpredictable long-term behavior even with a solution.

Despite the complexity and challenges, studying differential equations can provide insights into the underlying structure of the world around us.

Transcripts

Taking a quote from Stephen 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.

The aim is to give a big picture view of what this piece 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 too much beyond that.

Differential equations arise whenever it's easier

to describe change than absolute amounts.

It's easier to say why population sizes, for example,

grow or shrink than it is to describe why they 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, and force determines acceleration,

which is a statement about change.

These equations come in two different 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 differential equations are something we'll

be looking at more closely in the next video.

You often think of them as involving a whole continuum of values changing with time,

like the temperature at every point of 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.

And 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 example of differential equations.

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 Earth causes things

to accelerate downward at 9.8 meters per second per second.

Now unpack what that's really saying.

It means if you look at that object free from other forces,

and record its velocity at every second, these velocity vectors will accrue an

additional small downward component of 9.8 meters per second every second,

we call this constant 9.8 g for gravity.

This is enough to give us an example of a differential equation,

albeit a relatively simple one.

Focus on the y-coordinate as a function of time.

Its derivative gives the vertical component of velocity,

whose derivative in turn gives the vertical component of acceleration.

For compactness, let's write the first derivative

as y-dot and the second derivative as y-double-dot.

Our equation says that y-double-dot is equal to negative g, a simple constant.

This is one we can solve by integrating, which

is essentially working the question backwards.

First, to find velocity, you ask, what function has negative g as a derivative?

Well, it's negative g times t, or more specifically,

negative gt plus the initial velocity.

Notice that there are many functions with this particular derivative,

so you have an extra degree of freedom which is determined by an initial condition.

Now what function has this as a derivative?

It turns out to be negative one-half g times t squared plus that initial velocity

times t, and again we're free to add an additional constant without changing the

derivative, and that constant is determined by whatever the initial position is.

And there you go, we just solved a differential equation,

figuring out what a function is based on information about its rate of change.

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 pole on one of them 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, acceleration, is some function of position,

so you have this dance between two mutually interacting variables,

reminiscent of the dance between the two moving bodies which they describe.

This is reflective of the fact that often in differential equations,

the puzzles you face involve finding a function whose derivative and

or higher order derivatives are defined in terms of the function itself.

In physics it's most common to work with second order differential equations,

which means the highest derivative you find in this expression is a second derivative.

Higher order differential equations would be ones involving third derivatives,

fourth derivatives, and so on, puzzles with more intricate clues.

The sensation you get when really meditating on one of these

equations 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 t,

but they're 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.

To get a feel for what studying these can look like,

I want you to take some time digging into 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 2 pi times the square root of l over g,

where l is the length of the pendulum and g is the strength of gravity.

However, these formulas are actually lies, or rather,

approximations which only work in the realm of small angles.

If you were to go and measure an actual pendulum,

what you'd find is that as you pull it out farther,

the period is longer than what the high school physics formulas would suggest.

And when you pull it out really far, this value of theta

plotted versus time doesn't even look like a sine wave anymore.

To understand what's really going on, first things first,

let's set up the differential equation.

We'll measure the position of the pendulum's weight as a distance x along this arc,

and if the angle theta we care about is measured in radians,

we can write x as l times theta, where l is the length of the pendulum.

As usual, gravity pulls down with an acceleration of 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 theta.

So the component of gravity in the direction of motion

opposite this angle will be negative g times sine of 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.

This minus sign in the acceleration indicates that it's

always pointed in the opposite direction from displacement.

So what we have is that the second derivative of x,

the acceleration, is negative g times sine of theta.

As always, it's nice to do a quick gut check that our formula makes physical sense.

When theta is zero, sine of zero is zero, so there's

no acceleration in the direction of movement.

When theta is 90 degrees, sine of theta is 1, so the

acceleration is the same as it would be for freefall.

Alright, that checks out.

And because x is L times theta, that means the second

derivative of theta is negative g over L times sine of theta.

To be a little more realistic, let's add in a term to account for the air resistance,

which maybe we model as being proportional to the velocity.

We'll write this as negative mu times theta dot,

where mu is some constant that encapsulates all the air resistance

and friction and such that determines how quickly the pendulum loses energy.

Now this, my friends, is a particularly juicy differential equation.

It's not easy to solve, but it's not so hard that we can't

reasonably get some meaningful understanding out of it.

At first glance, you might think that the sine function you

see here 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 real pendulums don't oscillate with a sine wave pattern.

If that sounds odd, consider the fact that here,

the sine function is taking theta as an input,

but in the approximate solution you might see in a physics class,

theta itself is oscillating as the output of a sine function.

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 grapple with.

They're really freaking hard to solve.

In this case, if we remove that dampening term,

we can just barely write down an analytic solution, but it's hilariously complicated.

It involves all these functions you've probably never heard of,

written in terms of integrals and weird inverse integral problems.

When you step back, presumably the reason for finding a solution is to then be able

to make computations and build an understanding for whatever dynamics you're studying.

In this case, those questions have been punted off to figuring out how to compute,

and more importantly, understand, these new functions.

And more often, like if we add back in that dampening term,

there's not a known way to write down an exact analytic solution.

Well, for any hard problem you could just define a new function to be the answer of

that problem, heck, even name it after yourself if you want, but again,

that's pointless unless it leads you to being able to make computations and build

understanding.

So instead, in the study of differential equations, we often do a sort of short circuit,

and skip the actual solution part, since it's unattainable,

and go straight to building understanding and making computations from the

equations alone.

Let me walk through what that might look like with a pendulum.

What do you hold in your head, or what visualization can you get some software

to pull up for you, to understand the many possible ways that a pendulum,

governed by these laws, might evolve depending on its starting conditions?

You might be tempted to try imagining the graph of theta vs.

t, and somehow interpreting how this slope, position,

and curvature all interrelate with each other.

However, what will turn out to be both easier and more general is to

start by visualizing all possible states in a two-dimensional plane.

What I mean by the state of the pendulum is that you can describe it with two numbers,

the angle and the angular velocity.

You can freely change either one of those two values without necessarily

changing the other, but the acceleration is purely a function of those two values.

So each point of this two-dimensional plane fully

describes the pendulum at any given moment.

You might think of these as all possible initial conditions of that pendulum.

If you know the initial angle and the angular velocity,

that's enough to predict how the system will evolve as time moves forward.

If you haven't worked with them before, 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 is being represented.

Notice how at the start, as theta decreases, theta dot,

the y-coordinate, 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 always being represented by

the vertical component of our space.

It's important to remind yourself that this state space is an abstract thing,

and is distinct from the physical space where the pendulum itself lives and moves.

Since we're modeling this as losing some of its energy to air resistance,

this trajectory spirals inward, meaning the peak velocity and

peak displacement each go down 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 a vector, theta, theta dot.

Maybe you think of that as an arrow from the origin, or maybe you think of it 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 space.

Take a moment to interpret what this is saying.

The first component for this rate of change vector is theta dot,

which is also a coordinate in our space.

The higher up we are in the diagram, 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 itself.

In other words, the first derivative of our state vector is some function of

that vector itself, with most of the intricacy tied up in that second coordinate.

Doing the same at all points of this space will show

how that state tends to change from any position.

As is typical with vector fields, we artificially scale down the vectors when

we draw them to prevent clutter, but use color to loosely indicate magnitude.

Notice 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 really thinking of two separate values,

intertwined via this mutual effect they have on one another'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 wonderful visual way to

think about what solving the 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 field.

And again, keep in mind, this velocity is not the same thing as

the physical velocity of the pendulum, it's a more abstract rate of change,

encoding the rates of change for 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 the possible

ways the pendulum evolves from 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 enough initial velocity that it

fully rotates around several times before settling 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.

That's obvious when I call it the air resistance term,

but imagine that you saw these equations out of context,

not knowing that they described a pendulum.

It's not obvious just looking at them that increasing this value of mu

means the system as a whole tends towards some attracting state faster.

So getting some software to draw these vector fields for you

can be a great way to build 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 three-dimensional space evolve if

they act on each other with gravity, and if 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 in total,

and hence an 18-dimensional space of possible states.

It's a bizarre thought, isn't it?

A single point meandering through an 18-dimensional space that 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

the momenta of the three masses we see in ordinary, physical 3D space.

In practice, you can reduce the number of dimensions here by taking

advantage of the symmetries of your setup, but the point that more

degrees of freedom results in higher dimensional state spaces remains the same.

In math, we often call a space like this a phase space.

You'll hear me use that term broadly for spaces encoding all kinds of

states of changing systems, but you should know that in the context of physics,

especially Hamiltonian mechanics, the term is often reserved for a more special case,

namely a space whose axes represent position and momentum.

So a physicist would agree that the 18-dimensional space describing the

three-body problem is a phase space, but they might ask that we make a couple

of modifications to our pendulum setup for it to properly deserve the term.

For those of you who just watched the block collision video,

the planes we worked with there would 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, the author James Glick describes phase space as,

"One of the most powerful inventions of modern science".

One reason its powerful is that you can ask questions,

not just about a single initial condition but about 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 idea,

consider the question of stability.

The origin of our space corresponds to the pendulum standing still,

and so does this point over here, representing when the pendulum is perfectly

balanced upright.

These are the so-called fixed points of our system,

and one natural question to ask is whether or not they're stable, that is,

will tiny nudges to the system result in a state that tends back towards that

fixed point, or away from it?

Physical intuition for the pendulum makes the answer here kind of obvious,

but how would you think about stability just looking at the equations,

say if they arose in some completely different less intuitive context?

We'll go over how to compute the answers to questions like this in following videos,

and the intuition for the relevant computations are guided heavily by

the thought of looking at small regions in space around a fixed point,

and asking whether the flow tends to contract or expand.

And speaking of attraction and stability, let's take a brief side-step to talk about love.

The Strogatz quote that I mentioned earlier comes from a whimsical column in

the New York Times on the mathematics of modelling affection,

an example well worth pilfering to illustrate that we're not just talking

about physics here.

Imagine you've been flirting with someone, but there's been some frustrating

inconsistency to how mutual your affection seems,

and perhaps during a moment when you turn your attention towards physics

to keep your mind off the romantic turmoil, mulling over the broken-up

pendulum equations, you suddenly understand the on-again-off-again dynamics

of your flirtation.

You've noticed that your own affection tends 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 in the pendulum,

and you too 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,

involving a single variable, and another from, uh, chemistry, with two variables,

actually have a very similar structure, easier to recognize when you're looking at the

phase diagram.

Most notably, even though the equations are different,

for example there's no sine function in the romance equations,

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?

One way to do this is to essentially simulate what the universe would 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 in this phase diagram,

take a step based on the vector you're sitting on for a small time step, delta t.

Specifically, take a step equal to delta t times that vector.

As a reminder, in drawing these vector fields,

the magnitude for each vector has been artificially scaled down to prevent clutter.

When you do this repeatedly, your final location will be an approximation of theta t,

where t is the sum of all those time steps.

If you think about what's being shown right now, though,

and what that would imply for the pendulum's movement,

you'd probably agree that this is grossly inaccurate.

But that's only because the time step delta t of 0.5 is way too big.

If we turned it down, say to 0.01, you can get a much more accurate approximation,

it just takes more repeated steps is all.

In this case, computing theta of 10 requires 1000 little steps.

Luckily, we live in a world with computers, so repeating a simple task 1000

times is as simple as articulating that task with a programming language.

In fact, let's finish things off by writing a little

python program that computes theta of t for us.

What it has to do is make use of the differential equation,

which returns the second derivative of theta as a function of theta and theta dot.

You start off by defining two variables, theta and theta dot,

each in terms of some initial conditions.

In this case I'll have theta start at pi thirds,

which is 60 degrees, and theta dot start at 0.

Next, write a loop that corresponds to taking many little time steps

between 0 and time t, each of size delta t, which I'm setting here to be 0.01.

In each step of this 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 time steps, simply return the value of theta.

This is called solving a differential equation numerically.

Numerical methods can get way more sophisticated and intricate than this 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'll look at several methods for finding exact

solutions when it's possible, but one theme I'd like to focus on is how these

exact solutions can also help us to study the more general, unsolvable cases.

But it gets worse.

Just as there's 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 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 the 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 it should also be reassuring.

It gives some hope that the complexity we see in the world around us can be studied

somewhere in this math, and that it's not hidden away in the mismatch between model and

reality.