Foundations — Phase space — trajectories, phase portraits

This page assumes you know nothing yet. Before you can read the parent topic, every symbol it fires at you must first be earned. Below, each idea is built from the one before it — read top to bottom, no skipping.

0. The characters in the story

The parent note uses these symbols: $x$, $q$, $t$, $\dot{x}$, $\dot{q}$, $p$, $m$, $\omega$, $V$, $E$, $H$, $\partial$, the dot-over-a-letter, and the arrows of a "vector field". We define them one at a time.

1. Position — $x$ (and its cousin $q$)

The parent sometimes writes $q$ instead of $x$. They mean the same thing here: a generalized coordinate. The letter $q$ is just a more grown-up name that also works when "position" is an angle, a length, or anything that locates the system. For a swing, $q$ could be the angle $\theta$; for a bead on a wire, $q=x$.

2. Time and the dot — $t$ and $\dot{x}$

So $\dot{x}$ (read "$x$-dot") means "how fast $x$ is changing" — the velocity.

• $\dot{x}>0$: bead moving right.
• $\dot{x}<0$: bead moving left.
• $\dot{x}=0$: bead momentarily at rest (like a ball at the top of its throw).

A second dot, $\ddot{x}$, is "the rate of the rate" — how fast the velocity itself changes. That is acceleration. Two dots = acceleration.

3. Mass and momentum — $m$ and $p$

WHY the topic wants $p$ instead of $\dot{x}$: it makes the deep symmetry between "position" and "motion" appear (you'll see $\dot{q}$ and $\dot{p}$ sit in beautifully mirrored equations). But if that scares you, just read $p$ as "velocity in disguise."

4. Why we need TWO numbers — the second-order idea

Newton's law says $\text{force}=m\ddot{x}$. Notice the two dots: the law tells you the acceleration, not the position directly.

That pair $(x,\dot{x})$, or equivalently $(q,p)$, is called the state. This is the single most important reason phase space is drawn with two axes.

5. Potential energy $V$ and total energy $E$

Two facts you'll use constantly:

• A valley bottom (minimum of $V$) is a resting place you fall back toward → stable.
• A hilltop (maximum of $V$) is a resting place you fall away from → unstable.

$E=\underset{\text{kinetic}}{\underbrace{\frac{1}{2}m{\dot{x}}^2}}+\underset{\text{potential}}{\underbrace{V(q)}}.$

6. The energy bookkeeper — $H$

$H(q,p)=\frac{p^2}{2m}+V(q).$

WHY rewrite kinetic energy as $\frac{p^2}{2m}$? Because $p=m\dot{x}$, so $\dot{x}=p/m$ and $\frac{1}{2}m{\dot{x}}^2=\frac{1}{2}m(p/m{)}^2=\frac{p^2}{2m}$. Same energy, now expressed in the two state variables — exactly the pair phase space plots. This bookkeeping is the heart of Hamiltonian Mechanics.

7. The curly-$d$ — $\partial$ (partial derivative)

The parent note writes things like $\partial H/\partial p$. The curly $\partial$ scares beginners, so here it is from zero.

Here $\omega$ (Greek "omega") is just a fixed number setting how stiff/fast the spring is — bigger $\omega$, quicker wobble. You'll meet it fully in the Harmonic Oscillator.

8. A vector field — the "flow" the topic keeps mentioning

This single picture — arrows filling a plane, leaves drifting along them — is exactly what a phase portrait is. Everything in the parent note is dressing on top of this image.

The prerequisite map

Read it as: coordinates + time give velocity; velocity + mass give momentum; position and momentum together make the state; energy pieces build the Hamiltonian; the partial derivative turns the Hamiltonian into a field of arrows; the arrows over the state-plane are the phase portrait. That final box is the topic itself — and it opens the doors to Stability and Fixed Points, Liouville's Theorem, and the Pendulum.

Equipment checklist

Test yourself — you're ready when each reveal feels obvious.