Foundations — Hamilton's principle — least action

The parent note Hamilton's Principle throws a lot of symbols at you very fast: $q$, $\dot{q}$, $L$, $T$, $V$, $\int$, $\delta$, $\partial$, $\eta$, $\epsilon$. This page builds every single one from absolute zero, in the order they need each other. If you can read line one, you can read all of it.

1. Position, but flexible: the coordinate $q$

The picture. Think of a bead on a bent wire. You don't need $x$ and $y$ — one number "how far along the wire" already says everything. That one number is $q$.

Why the topic needs it. Forces live in $x,y,z$ and you must chop them into components. A well-chosen $q$ (like the pendulum's angle $\theta$) absorbs the geometry so you never decompose anything. That is the entire practical payoff of the whole subject.

2. Speed of the coordinate: $\dot{q}$ and the dot

The picture. If $q$ is the position of a runner on a track, $\dot{q}$ is the speedometer reading, and $\ddot{q}$ is whether they're accelerating or braking.

$\dot{q}=\frac{dq}{dt},\ddot{q}=\frac{d^{2}q}{d{t}^2}.$

Why the notation. Physicists put the dot specifically for time derivatives, freeing the prime ${}^{\prime }$ for space derivatives. In $L(q,\dot{q},t)$ the system's "state" needs both where it is ($q$) and how fast it moves ($\dot{q}$) — position alone can't tell you its kinetic energy.

3. Two kinds of change: $T$ and $V$ (energy)

The picture. A ball rolling in a valley. Deep in the valley it moves fast ⇒ big $T$, small $V$. At the top of the slope it pauses ⇒ big $V$, small $T$. As it rolls, energy sloshes back and forth between the two buckets.

Why the topic needs both. The whole machine is built from the difference $T-V$. We need each bucket separately before we can subtract them.

4. The star of the show: the Lagrangian $L$

The picture. At every tick of the clock the ball has some $T$ and some $V$; subtract them to get one number $L$. As the ball moves, $L$ traces out a curve over time.

Why $L$ is written $L(q,\dot{q},t)$. It depends on where you are ($q$, through $V$), how fast you move ($\dot{q}$, through $T$), and possibly the clock ($t$, if the setup changes in time). Three slots — remember them, they matter in step 8.

5. Adding up over the journey: the integral $\int$

The picture. Slice the journey into millions of tiny time-steps. In each step, $L$ has some value; multiply by the tiny duration $dt$ (a thin strip). Stack all the strips ⇒ the total shaded area under the $L$-versus-$t$ curve.

Why the topic needs it. A path isn't judged at one instant — it's judged over its whole duration. Integration is precisely the tool that collapses "value at every instant" into "one total score."

6. The score itself: the action $S$ and the word functional

The picture. Feed in a path — any wiggly curve from start to end — and out pops a number. A different path gives a different number. Note the square brackets $S[q]$: they warn you the input is a whole function, not just a value.

Why "functional" and not "function"? A plain function eats a number and gives a number ($f(3)=9$). A functional eats a function. This is one level up, and it's the reason we need the Calculus of Variations — ordinary calculus differentiates with respect to a number; here we must "differentiate with respect to a whole curve."

7. The tiny nudge: $\eta$, $\epsilon$, and the wiggle

To find the best path we compare it against nearby paths. We make a nearby path by adding a small bump:

$q(t)⟶q(t)+\epsilon \eta (t).$

The picture. Take the true path and gently pluck it in the middle like a guitar string, holding both ends fixed. The pluck shape is $\eta$; how hard you pluck is $\epsilon$. Because the ends are nailed down, every trial path still starts and finishes at the same two points.

Why pinned at the ends. Hamilton's principle asks "of all paths with the same start and finish, which is best?" We're not allowed to move the endpoints, so the nudge must vanish there. (This pinning is exactly what kills a leftover term later — see Euler–Lagrange equation.)

8. Two flavours of derivative: $\partial$ vs $\frac{d}{dt}$

The picture. $L$ is like a dashboard with three dials: $q$, $\dot{q}$, $t$. A partial derivative turns one dial and watches $L$. A total time-derivative lets the clock run and all the dials move at once (because $q$ and $\dot{q}$ both drift as time passes).

Why the topic needs both. The Euler–Lagrange equation $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial q}=0$ mixes them: first take a partial (isolate the $\dot{q}$-slot), then a total time derivative (let it evolve). Confusing the two is the single most common slip.

9. The stationary condition: $\delta S=0$

The picture. Imagine the action-score as the height of a landscape, where each point on the ground is a whole path. The true path sits at a flat point — a valley bottom, a hilltop, or a saddle — where the ground is level in every direction. A ball placed there wouldn't roll.

How the foundations feed the topic

Each foundation is a prerequisite for Euler–Lagrange equation, which in turn powers the parent Hamilton's Principle. From there you can branch to Lagrangian Mechanics, Noether's Theorem, Hamiltonian Mechanics, Fermat's Principle, and see how it all reduces to Newton's Laws.

Equipment checklist

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