Foundations — Hamilton-Jacobi equation

This page builds every symbol the parent note leans on, starting from a smart-12-year-old level. Read it before the Hamilton-Jacobi equation proper. Each entry gives plain meaning → the picture → why the topic needs it.

1. Position $q$ and momentum $p$ — the two numbers that fully describe a system

The picture. Draw a flat sheet. The horizontal axis is $q$ (where), the vertical axis is $p$ (how fast, mass-weighted). One dot on this sheet = one complete snapshot of the system. This sheet is called phase space.

Why the topic needs it. Hamilton-Jacobi lives entirely in phase space. It transforms the pair $(q,p)$ into a new pair $(Q,P)$. So we must first be comfortable that a point — not a curve — is the state, and that motion is that point sliding around. (More in Hamilton's Equations.)

For $n$ moving parts we have $q_1,\dots ,q_n$ and $p_1,\dots ,p_n$. The subscript $i$ just labels which part. The parent's ${\sum }_i$ ("sum over $i$") means "add up the contribution from every part." When a system moves in a straight line we often call that one coordinate $x$ instead of $q$ — same idea, different letter.

2. The dot: $\dot{q}$ means "rate of change"

The picture. On the phase-space sheet, $\dot{q}$ and $\dot{p}$ are the arrow of motion attached to the dot — the direction and speed the state is drifting. If $\dot{q}=0$ and $\dot{p}=0$, the dot is frozen.

3. Derivative $\frac{d}{dt}$ and partial derivative $\frac{\partial }{\partial q}$ — WHY two kinds

A derivative answers: "if I nudge the input a tiny bit, how much does the output change?" Geometrically it is the slope of a curve.

The picture. Think of a hillside whose height is $f(q,t)$.

The partial $\partial f/\partial q$ is the steepness if you walk due east (only $q$ changes). The total $df/dt$ is the steepness along the actual path you walk, which curves through both directions at once.

4. The Lagrangian $L$ and the action — "cost of a path"

Before the formula, two words of energy.

The picture. Imagine a path drawn in space over time. At each instant it has some $L$ value. Lay all those values side by side and shade the area underneath — that shaded area is the action for that path. The true motion is the path whose action is stationary (see Action and Hamilton's Principle).

Why the topic needs it. The parent's punchline is that Hamilton's principal function is the action: $S=\int Ldt+\text{const}$. So $S$ is not an abstract fudge — it is the accumulated "cost" along the real trajectory. Knowing what $L$ and the action mean makes that identity feel inevitable rather than magical.

5. The Hamiltonian $H$ — total energy as a function of $(q,p)$

The picture. Above the phase-space sheet, imagine a landscape whose height at each point $(q,p)$ is the energy $H$. A system with fixed energy $E$ is a hiker forced to walk along one contour line $H=E$ of that landscape.

Why the topic needs it. $H$ is the engine of the HJ equation: you take the Hamiltonian and everywhere you see a momentum $p_i$, you replace it with the derivative $\partial S/\partial q_i$. Understanding $H$ as "energy in $(q,p)$ language" is what lets that substitution mean something.

6. Legendre transform — the bridge $L\leftrightarrow H$

First, where momentum comes from in the Lagrangian world:

Why the topic needs it. In §3 of the parent, the chain $\sum p_i{\dot{q}}_i-H=L$ is exactly this relation run backwards. It is the hinge that turns "$dS/dt$ equals $\sum p\dot{q}-H$" into the clean statement "$dS/dt=L$." You don't need to re-derive it here — just recognize the shape $p\dot{q}-H=L$ when it appears, knowing $p_i=\partial L/\partial {\dot{q}}_i$ is what makes it true.

7. Canonical transformation and the generating function

The picture. Take the phase-space sheet and repaint the grid lines: new axes $Q$ and $P$ drawn over the old ones. A canonical transformation is a special repainting that does not distort the underlying flow rules — like changing from Cartesian to polar graph paper without breaking the geometry.

Why type 2? We want the new momenta $P$ to be constants. A type-2 function already treats $P$ as an independent input, so it is the natural tool to design constant new momenta. (Full menagerie in Canonical Transformations.)

8. The new Hamiltonian $K$ (the "Kamiltonian")

9. Separation of variables and "complete integral" constants

First, the constant that shows up when energy is conserved:

The picture. Peeling $-Et$ off $S$ is like separating a receipt into a fixed "per-second rate $E$" times elapsed time, plus a purely spatial remainder $W(q)$. Only possible when the energy rate $E$ is constant — i.e. energy conserved.

Why the topic needs it. These constants are not leftovers — they are the new momenta $P$ that make the whole canonical transformation exist. Without them there is no map. (This becomes Action-Angle Variables later.)

Prerequisite map

Everything on the left is what THIS page builds; the single node Hamilton-Jacobi equation is the parent topic it all feeds. See also the bridge to wave mechanics in Schrodinger Equation.

Equipment checklist

Hide the right side and test yourself. You are ready when each is instant.