Intuition The one core ideaOut of all imaginable routes a system could take between a fixed start and a fixed finish, nature picks the single route where one running total — the action — stops changing under any tiny nudge of the path. To even read that sentence in symbols, you first need to speak "function", "derivative", "integral", and "a number that eats a whole function"; this page builds every one of those from nothing.
This page is the toolbox for the parent derivation . Nothing here is assumed — we start below zero.
Picture a bead sliding along a wire. At each moment in time it sits somewhere . That "somewhere" is a single number if the wire is a straight track.
t q
t time ) is the clock reading. Picture: a number on a stopwatch that only goes up.q generalized coordinate ) is "where the system is right now", boiled down to a number. For a bead on a wire, q q
Why "generalized"? So the same machinery works whether the natural label is a length, an angle, or anything else. The topic needs ONE symbol that means "the state" regardless of the setup.
Look at the blue curve above: the horizontal axis is time t q path .
q ( t ) function is a machine: feed it a time t q q ( t ) whole blue curve in the figure above — the complete history of the bead, not just one moment.
Intuition Two different things wearing the letter q
The function q q ( ⋅ ) whole path — the entire blue curve.
The value q ( t ) single number the path returns at one instant t
Intuition Why the parent obsesses over "the whole path"Newton describes motion instant-by-instant (force now → acceleration now ). Hamilton's principle instead judges the entire curve at once . So the fundamental object we manipulate is a whole function q
Related deeper ideas live in Newton's Second Law and Lagrangian Mechanics .
The bead isn't just somewhere ; it's moving . How fast?
Definition The dot means "rate of change in time"q ˙ velocity : how fast q derivative of q t d t d q steepness (slope) of the blue curve at a point. Steep curve = moving fast; flat curve = momentarily still.
Intuition WHY a derivative and not just a difference?We want the instantaneous speed — the slope exactly at time t calculus limit delivers.
The orange line is the tangent — its slope is q ˙
Definition Partial derivative∂ / ∂ ( ⋅ ) partial derivative ∂ x ∂ f only x f ∂
q ˙ q Why it feels right: in a formula like 2 1 m q ˙ 2 − V ( q ) q q ˙ separate slots when taking partial derivatives (nudge one, freeze the other).
The fix: for a given path , q ˙ q
Definition Kinetic energyT T motion . For a mass m q ˙ T = 2 1 m q ˙ 2 . ≥ 0 q ˙ 2 m mass — "how much stuff", how hard to shove.
Definition Potential energyV V position — energy you'd get back by letting the system fall/relax. Depends on where you are: V = V ( q ) V V
Intuition Why the topic needs BOTH separatelyThe action combines them as T − V minus potential). Newton's force is hidden inside V F = − d q d V T V F = ma
The green valley is V ( q ) downhill , i.e. toward smaller V F = − d V / d q
minus , not plus?
This is the parent's central non-obvious choice. Justification is not a guess — plugging T − V T + V
∫ t 1 t 2 ( ⋯ ) d t ∫ "add up continuously" . ∫ t 1 t 2 L d t d t L t 1 t 2 area under the L t .
Intuition Why an integral and not a plain sum?Time flows smoothly, so we need infinitely many infinitely-thin slices. The integral is exactly "a sum that has gone continuous". It answers: "what is the total accumulated L
The shaded area is the integral. Change the path q ( t ) L ( t )
S [ q ] S [ q ] = ∫ t 1 t 2 L d t . functional is a machine that eats an entire function q one number (an area). The square brackets S [ q ]
Intuition Ordinary function vs functional
Function q ( t ) →
Functional S [ q ] whole curve in → stationary ?
η ( t ) ε q ε ( t ) = q ( t ) + ε η ( t ) .
η ( t ) wiggle-shape we're free to choose. It must be continuously differentiable (C 1 η ˙ ε volume knob for how big the wiggle is.Endpoints stay pinned: η ( t 1 ) = η ( t 2 ) = 0
δ S = 0 δ S first-order change in S ε δ S = 0 nudging the path a little doesn't change the action to first order .
Picture: the bottom of a valley — flat there, so a small step sideways doesn't change your height (to first order).
The blue curve is the true path; the dashed orange curves are wiggled versions q + ε η S ε ε = 0 flat — that flatness is δ S = 0
path is a function q of t
instantaneous value q at time t
derivative q-dot the slope
Lagrangian L equals T minus V
integral adds L over the trip
vary the path with eta and epsilon
stationary delta S equals zero
Hamilton principle derivation
Related destinations once you have these tools: Hamiltonian Mechanics , Noether's Theorem , Fermat's Principle , Feynman Path Integral .
Does the function q The whole curve — the complete history over time.
Does the value q ( t ) A single number — the position at the one instant t
What picture is the derivative q ˙ The slope (steepness) of the position-vs-time curve at an instant.
What does a partial derivative ∂ f / ∂ x The rate of change of f x
Why is T = 2 1 m q ˙ 2 ≥ 0 Because q ˙ 2
In one line, what is the force in terms of V Minus the slope of the potential, F = − d V / d q
Is L T + V T − V T − V
What does the integral ∫ L d t The area under the L t
How is a functional different from a function? A functional eats a whole function and returns one number; a function eats a number.
What smoothness must the wiggle η C 1
What must η η ( t 1 ) = η ( t 2 ) = 0
What does δ S = 0 The action is flat (stationary) under tiny path nudges — like the bottom of a valley.
Does "stationary" guarantee a minimum? No — it can be a minimum, maximum, or saddle.
