Differential equations, a tourist's guide | DE1

Differential equations, a tourist’s guide | DE1

Source: Differential equations, a tourist’s guide | DE1, 3Blue1Brown, 27:16, uploaded 2019-03-31, category Mathematics, playlist index 1325.

Steven Strogatz says that, since Newton, people have learned to express the laws of physics in the language of differential equations. 3Blue1Brown takes that as an invitation to look around rather than start with a formal definition. The tour assumes basic calculus and later asks for a little linear algebra. Its first claim is simple: a differential equation becomes useful when the reason something changes is easier to describe than its current amount.

Change before amount

Population growth gives the ordinary example. A model can say why a population grows or shrinks even when the exact population at a particular moment is hard to explain. The same structure can describe a changing affection. In Newtonian mechanics, force determines acceleration, and acceleration describes how velocity changes.

The video separates ordinary differential equations from partial differential equations. An ordinary differential equation, or ODE, follows a finite collection of values that depend on one input, often time. A partial differential equation describes a quantity that varies across several inputs, such as the temperature at every point in a solid or the velocity of a fluid throughout space. The distinction is about the number of independent inputs, whilst time remains the familiar setting for both.

Falling bodies and initial conditions

The vertical motion of a thrown object supplies a warm-up. Near Earth’s surface, gravity gives the object a downward acceleration of $9.8$ metres per second per second, written as $g$ . If $y$ is the object’s height, then its first derivative is vertical velocity and its second derivative is vertical acceleration:

\ddot{y} = -g.

Integration works backwards from the acceleration. A function whose derivative is $-g$ has the form

\dot{y}(t) = -gt + v_0,

where $v_0$ is the initial velocity. Integrating again gives

y(t) = -\frac{1}{2}gt^2 + v_0t + y_0,

where $y_0$ is the initial position. The derivative leaves room for an added constant, so each integration introduces a value that the initial conditions must fix. Solving the equation means recovering a function from information about its rates of change.

The problem grows when the forces depend on position. Gravitational motion between planets, stars, or moons gives an acceleration whose direction follows the other body and whose strength falls with the square of the distance. Position changes into velocity, velocity changes into acceleration, and acceleration depends on position again. A differential equation can therefore ask for a function whose derivatives are defined in terms of the function itself.

Physics often uses second-order equations, although higher-order equations bring in third and fourth derivatives as well. The source describes the task as finding infinitely many values, one for every moment, under a rule that ties those values to their own changes and to changes of those changes.

The pendulum and the failure of the schoolbook formula

The pendulum gives the first difficult case. Introductory physics often treats its angle $\theta$ as a sine wave with period

T = 2\pi\sqrt{\frac{L}{g}},

where $L$ is the pendulum’s length. That formula describes small angles. As the pendulum swings farther out, its period becomes longer than the schoolbook result predicts, and the graph of $\theta$ over time loses its sine-wave shape.

To model the motion, the video measures distance along the pendulum’s arc as $x=L\theta$ , with $\theta$ in radians. Gravity supplies an acceleration of $g$ downward, although the pendulum can move only along the arc. The component in the direction of motion is $-g\sin(\theta)$ , since the restoring acceleration points against the displacement. The resulting equation is

\ddot{\theta} = -\frac{g}{L}\sin(\theta).

The formula passes two physical checks. At $\theta=0$ , the tangential acceleration is zero. At $90$ degrees, it has the magnitude of freefall. Adding air resistance and friction gives the damped model

\ddot{\theta} = -\frac{g}{L}\sin(\theta) - \mu\dot{\theta},

where $\mu$ describes how quickly the pendulum loses energy. The video’s description corrects an on-screen equation at 6:27: the coefficient is $g/L$ , rather than $L/g$ .

The sine in this equation does not mean that the pendulum itself follows a sine wave. Here sine takes the angle as its input and determines the acceleration. The familiar sine-shaped motion comes from the small-angle approximation that replaces $\sin(\theta)$ with $\theta$ . The full equation is harder to solve, even though the pendulum looks simple. Removing damping permits an analytic expression, although the expression introduces complicated functions defined through integrals. Restoring damping leaves the source with no known exact analytic expression of the usual kind.

The distinction between defining an answer and understanding it matters here. One could name a new function as the answer to every difficult equation. That name would help little unless it allowed useful computation or explained the system’s behaviour. Differential equations therefore often get studied directly, with the solution step replaced by visual analysis and approximation.

State space and the vector field

A graph of $\theta$ against time gives one trajectory. It becomes more useful to draw every possible state of the pendulum in a plane. A state needs two numbers: the angle $\theta$ and the angular velocity $\dot{\theta}$ . Any point in this plane describes the pendulum at one moment, and the two coordinates can vary independently. Once they are known, the equation determines the acceleration and therefore the future motion.

This plane is an abstract state space. It is separate from the physical space in which the pendulum swings. With air resistance, a typical trajectory spirals towards the origin. The angle and angular velocity both fall as the pendulum loses energy. A large initial velocity can send the point to the right across the diagram as the pendulum makes several full rotations before it settles into a smaller oscillation. Increasing $\mu$ makes the spiral contract faster.

The differential equation supplies a vector at every point in the plane. Treat the state as the vector $(\theta,\dot{\theta})$ . Its derivative is

\frac{d}{dt} \begin{pmatrix} \theta \\ \dot{\theta} \end{pmatrix} \;=\; \begin{pmatrix} \dot{\theta} \\ \ddot{\theta} \end{pmatrix}.

The first component says how the point moves horizontally: a positive angular velocity moves it towards larger $\theta$ . The second component comes from the pendulum equation and depends on the current state. Drawing that vector at every point produces a vector field. A solution is a trajectory whose velocity agrees with the field at every moment.

This also turns one second-order equation into two first-order equations. The substitution $v=\dot{\theta}$ gives

\dot{\theta}=v, \qquad \dot{v}=-\frac{g}{L}\sin(\theta)-\mu v.

The trick applies to any system of ODEs, which makes vector fields a general way to study equations that may have no manageable closed-form solution. Software scales the arrows down so the diagram stays legible and uses colour to suggest their magnitude. Once the field is visible, changing a parameter such as $\mu$ reveals a change in the whole system rather than only in one calculated trajectory.

Phase space and stability

The pendulum’s plane is small enough to draw. The three-body problem shows why the same idea soon becomes abstract. Three masses moving through three-dimensional space have three position coordinates and three momentum coordinates each, giving an 18-dimensional space of possible states. A single point moves through that space as time passes, while its coordinates encode everything needed to reconstruct the three bodies in ordinary space.

The source uses phase space as a broad name for a space that records the states of a changing system. In physics, especially Hamiltonian mechanics, the term often has the narrower meaning of a space whose axes represent position and momentum. The broader mathematical usage and the stricter physical usage overlap, although the exact boundary depends on the context.

Phase space lets us study a whole family of initial conditions at once. The collection of trajectories becomes a flow. The origin of the pendulum diagram represents the motionless hanging pendulum. The point for the perfectly upright pendulum is another fixed point. Stability asks what happens after a small nudge: does the nearby flow contract towards the fixed point or expand away from it? The equations can answer this even when the physical system gives no helpful intuition. The relevant calculation examines a small region around the fixed point and measures how the flow changes there.

A phase diagram for affection

The Strogatz quotation comes from a New York Times column about modelling affection. The video uses it to move the same mathematical structure into a deliberately playful setting. Imagine that one person’s affection grows when the other seems interested and falls when the other becomes cold. The second person’s affection follows the opposite pattern, rising when the first person seems uninterested and falling when the first person becomes too keen.

The two feelings form a two-dimensional state. Their phase diagram resembles the central part of the pendulum diagram, so the pair can move through an endless cycle of attraction and repulsion. If the second person’s feelings slow down as they become too vulnerable, the model gains a damping term. The trajectory then spirals towards mutual ambivalence.

The equations differ from the pendulum equation, including the absence of a sine term, yet their phase diagrams expose the same kind of structure. The source’s point is methodological. A technique developed around one physical example can transfer to another system because the geometry of the flow carries information that the surface story conceals. The romance model remains a constructed analogy. Its mathematical similarity to a pendulum does not establish a law of human relationships.

Numerical steps

Phase diagrams show how a system behaves. To compute a particular trajectory, the video imitates the system in small finite steps. At a point in the phase plane, take the vector attached to that point, multiply it by a small time step $\Delta t$ , and add the result to the current state. Repeating this process approximates the position after the sum of those time steps.

A step of $\Delta t=0.5$ produces a visibly poor pendulum trajectory. Reducing it to $0.01$ gives a much closer approximation, although reaching $t=10$ then requires 1,000 updates. A short Python loop can handle that repetition. The example starts with

\theta(0)=\frac{\pi}{3}, \qquad \dot{\theta}(0)=0,

and updates the two state values at every step:

\theta \leftarrow \theta + \dot{\theta}\,\Delta t,

\dot{\theta} \leftarrow \dot{\theta} + \ddot{\theta}\,\Delta t.

The acceleration comes from the differential equation. After the loop, the current value of $\theta$ approximates the angle at the requested time. This is the basic idea of numerical solution. More advanced methods choose their steps with greater care to balance accuracy and efficiency, although the small loop already shows how a computer can study an equation that resists exact solution.

Exact solutions and the limits of prediction

Exact methods remain useful when they exist, and the later videos in the series return to them. Exact solutions can also help researchers understand systems whose general cases have to be approximated. A formula alone cannot remove the limits of the model or of prediction.

Chaos theory adds a second limit. Some systems react wildly to small changes in their initial conditions, including changes that arise from imperfect measurement. The three-body problem contains the seeds of this behaviour. A precise solution may therefore fail to provide useful long-term prediction because a tiny error in the starting state sends the trajectory somewhere else.

The video returns to Strogatz’s opening claim with that tension in view. Differential equations give a language for the complexity of the world, although the language contains problems that resist exact solution and systems whose long-term paths remain sensitive to measurement. That difficulty gives the language its reach. The complexity stays available to study inside the mathematics rather than disappearing into the gap between a model and the world it describes.

Limits

This note follows the complete English caption track, the video’s chapters, and its description. The equations and examples belong to 3Blue1Brown’s introductory presentation. The caption track does not provide a proof of the claims about analytic solvability, phase-space terminology, or chaos, so those claims remain attached to the source. The romance example is a model chosen for analogy, and the video gives no evidence that it predicts real relationships. The description supplies one explicit equation correction, which is preserved above.