Foundations — Hamilton's equations of motion

This page assumes nothing. If the parent note used a symbol, we build it here from a picture first. Read top to bottom; each block earns the next.

0. What all these letters even are

Before any formula, here is the cast of characters you will meet. We define each one properly below — this is just so you know who is coming.

Now we build every one of these, in order, from zero.

1. A number that can move: the variable $q$

Imagine a bead sliding on a wire, or a pendulum swinging. To say where it is we need one number. Call it $q$.

• If the bead is on a straight wire, $q$ is its distance from a marked zero — like a ruler reading.
• If it's a pendulum, $q$ could be the angle from straight-down.

Why does the topic need it? Every law of motion is ultimately a statement about how $q$ changes over time. No $q$, nothing to describe.

If there are several moving parts, we write $q_1,q_2,\dots$ The little number is called a subscript (an index). We write $q_i$ to mean "the $i$-th one" — a placeholder standing for all of them at once.

2. The rate of change: the dot, $\dot{q}$

Watch the bead. In a tiny sliver of time it moves a tiny distance. Speed is "distance moved ÷ time taken." As we shrink the time sliver to nearly zero, this ratio settles onto a single number: the instantaneous velocity.

What does it look like? Plot $q$ upward and time $t$ sideways. The bead's history is a curve. The dot $\dot{q}$ is the steepness (slope) of that curve at each instant — steep means fast, flat means momentarily still.

Two dots, $\ddot{q}$, mean the slope of the slope — how fast the velocity changes, i.e. acceleration.

3. Slope of a landscape, in more than one direction: $\partial$

Now suppose a quantity depends on two things at once — say the height $H$ of a hilly landscape depends on how far east ($q$) and how far north ($p$) you stand: $H(q,p)$.

If I ask "how steep is the hill?", I must ask "steep in which direction?"

What it looks like: stand on the hill. Face due east and note the slope under your feet — that's $\partial H/\partial q$. Turn to face north without moving — that new slope is $\partial H/\partial p$. Same spot, two different steepnesses.

Why the topic needs it. Hamilton's equations say the velocity of the system equals a slope of $H$ in one direction, and the change of momentum equals minus a slope in the other direction. Without "slope in a chosen direction," you cannot even write the equations.

4. The push a system carries: momentum $p$

For a plain particle of mass $m$, momentum is $p=m\dot{q}$ — heavy things and fast things carry more "push." But Hamilton needs a more general momentum that works for angles, fields, weird coordinates.

You don't understand $L$ yet — that's the next section. For now hold this thought: momentum is a slope of $L$ in the velocity direction. That's why we needed $\partial$ first.

Why we bother with $p$ at all. Hamilton's whole trick is to stop using velocity $\dot{q}$ and use this push $p$ instead. The payoff (built on the parent page) is a perfectly symmetric pair of rules for $q$ and $p$.

5. Adding up over all parts: $\sum$

If the system has several coordinates, we often add one term per coordinate.

For a system with one moving part, ${\sum }_i$ has a single term and you can mentally ignore it. It only earns its keep when there are many parts.

6. The Lagrangian $L$ — the thing we start from

The parent page begins with $L$. Here is what it is, from zero.

Why a difference $T-V$ and not the sum? This is subtle and you should not memorise it as "energy." $L$ is a bookkeeping device: Nature moves a system so that a running total of $L$ over the whole path is as small as it can be (the principle of least action, from Lagrangian Mechanics). The difference $T-V$ is exactly the combination that makes this work. We accept it here as our given starting material.

What it depends on. $L$ is a function of position, velocity, and possibly time: $L(q,\dot{q},t)$. That "and time" matters later — if $L$ secretly changes with $t$ (e.g. someone is shaking the apparatus), energy need not be conserved.

7. The Hamiltonian $H$ — where we're heading

$H$ is built from $L$ by a swap called the Legendre transform (its own deep-dive: Legendre Transform). We only preview it here.

The picture that makes $H$ special: think of $H(q,p)$ as a landscape over the flat map whose east-axis is $q$ and north-axis is $p$. That map is called phase space (Phase Space and Liouville's Theorem). The system's whole life is a single point wandering on this map, and $H$'s slopes tell it where to go next.

8. How the pieces fit together

Read it as a supply chain: position and velocity feed $L$; the partial derivative of $L$ in the velocity direction makes momentum $p$; $p$, the sum, and $L$ assemble into $H$; and the slopes of $H$ are the equations of motion that the whole topic is about.

Equipment checklist

Test yourself — cover the right side. If any answer surprises you, reread that section before the parent page.