Foundations — Hamiltonian — definition H = Σpᵢq̇ᵢ − L

This page assumes nothing. Every squiggle you will meet in the parent note is built here from the ground up, in an order where each item leans only on the ones before it.

1. A "system" and its state

Imagine a bead sliding on a wire, or a swinging pendulum. To know everything about the motion at one instant, you need two kinds of fact:

• Where is it? — a position.
• How is it moving? — a velocity (later swapped for momentum).

Everything below is machinery for writing these two facts precisely and turning one description into the other.

2. Generalized coordinate $q_i$

The picture: for the pendulum in figure s01, one number — the angle $\theta$ from straight-down — completely fixes the bob's location. So $\theta$ is a generalized coordinate. We don't need $x$ and $y$ separately; the wire/rod already constrains them.

• The letter $q$ is the traditional symbol.
• The little subscript $i$ is just a label — a name tag. If a system needs three numbers to describe it, we call them $q_1,q_2,q_3$. The symbol $q_i$ means "the $i$-th one, whichever you like".

3. The dot: ${\dot{q}}_i$ means rate of change

Where the dot comes from — the derivative. Suppose the coordinate is $q$ at time $t$ and $q+\Delta q$ a tiny moment $\Delta t$ later. The average speed of change is $\Delta q/\Delta t$. Shrink $\Delta t$ toward zero and this ratio settles on one number — the instantaneous rate. That limiting number is what the dot denotes: $\dot{q}={\lim }_{\Delta t\to 0}\frac{\Delta q}{\Delta t}=\frac{dq}{dt}.$

Why this tool and not another? We need to know how fast the coordinate moves right now, not on average over a whole second. Only the limit — the derivative — gives an honest "right now" answer. That is precisely the question the derivative was invented to answer.

The picture: plot $q$ against time. The dot is the steepness (slope) of that curve. Steep upward $\Rightarrow$ large positive $\dot{q}$; flat $\Rightarrow$ $\dot{q}=0$; sloping down $\Rightarrow$ $\dot{q}$ negative.

4. The partial derivative $\frac{\partial }{\partial {\dot{q}}_i}$

The parent note writes things like $\partial L/\partial {\dot{q}}_i$. Here is that symbol, from zero.

Why we need it: the Lagrangian $L$ depends on $q$, on $\dot{q}$, and on time $t$ simultaneously. To find the momentum we must isolate the response of $L$ to velocity alone. The partial derivative is the only tool that answers "response to one input, others frozen".

The picture: think of $L$ as the height of a landscape over a floor whose two directions are $q$ (east) and $\dot{q}$ (north). The partial $\partial L/\partial \dot{q}$ is the slope you feel walking due north, ignoring any east–west tilt.

5. The Lagrangian $L$

• $T$ is big when things move fast: e.g. $T=\frac{1}{2}m{\dot{x}}^2$. Depends on velocity.
• $V$ is stored energy that depends on where you are (height, spring stretch): $V(q)$. Usually velocity-free.

Why $T-V$ and not $T+V$? That minus sign is what makes the Euler–Lagrange machinery (below) reproduce Newton's laws — take it on faith here; it is developed in Lagrangian Mechanics. For this topic, $L$ is simply the raw material the Hamiltonian is built from.

6. Conjugate momentum $p_i$

The picture (reusing s03): $p_i$ is exactly the north-slope of the $L$-landscape. Steeper dependence of $L$ on velocity $\Rightarrow$ larger momentum.

Sanity check with the everyday case. For $L=\frac{1}{2}m{\dot{x}}^2-V(x)$, nudging $\dot{x}$ gives $p=\partial L/\partial \dot{x}=m\dot{x}$ — ordinary "mass times velocity". So the fancy definition contains the familiar one as a special case.

7. The sum symbol ${\sum }_i$

Why we need it: a system may have many coordinates. Rather than write a long chain of plus signs, $\sum$ packs "one term per coordinate, all added" into one compact symbol.

8. Energy words: $T$, $V$, $E$

• $T$ — kinetic energy, energy a thing has because it is moving. Larger for faster or heavier objects.
• $V$ — potential energy, energy stored by position: a raised weight, a stretched spring.
• $E=T+V$ — total energy. In a system with no friction/driving, $E$ stays constant as motion sloshes energy back and forth between $T$ and $V$.

The parent note's punchline — "$H$ often equals $E$" — only makes sense once you hold these three straight. $E$ is the physical energy; $H$ is a constructed quantity that sometimes, not always, coincides with it.

9. Putting the symbols to work (the Hamiltonian)

Now every piece of $H={\sum }_i{p}_i{\dot{q}}_i-L$ is defined:

The parent note then trades the velocity description for the momentum description. Everything downstream — Legendre Transform, Hamilton's Canonical Equations, Phase Space, Poisson Brackets, Conservation Laws & Noether's Theorem — is built on exactly these atoms.

10. How the foundations feed the topic

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, re-read that section.