M081 · Action / boundary value

Principle of Stationary Action

Drag a candidate path while fixing its endpoints. Compare action, first variation and equation residual in free motion, uniform gravity and harmonic motion; move from a minimum to a saddle and test compatible or impossible conjugate-time boundaries.

Reviewed time $\text{Pending}$

Elapsed interval $\text{Pending}$

Candidate position $\text{Pending}$

Candidate velocity $\text{Pending}$

Candidate action $\text{Pending}$

Base action $\text{Pending}$

Action change from base $\text{Pending}$

Selected-mode slope $\text{Pending}$

Selected-mode curvature $\text{Pending}$

Six-mode gradient norm $\text{Pending}$

Newton residual RMS $\text{Pending}$

Quadrature discrepancy $\text{Pending}$

Negative-curvature directions $\text{Pending}$

Reference classification $\text{Pending}$

Endpoint compatibility $\text{Pending}$

Projected stationarity $\text{Pending}$

OBSERVE / VARY / VERIFY

Must the physical path have the smallest action?

System: free / gravity / harmonic

\mathcal M

2.00\,

Particle mass

m

1.00\,kg

Harmonic angular frequency

\omega

1.00\,rad/s

Uniform gravitational acceleration

g

1.00\,m/s^2

Interval scale

T/T_0

0.70\,

Fixed initial position

q_0

0.50\,m

Fixed final position

q_T

-0.20\,m

Scanned sine mode

n

1.00\,

Sine amplitude 1

a_1

0.60\,m

Sine amplitude 2

a_2

-0.20\,m

Sine amplitude 3

a_3

0.00\,m

Sine amplitude 4

a_4

0.00\,m

Sine amplitude 5

a_5

0.00\,m

Sine amplitude 6

a_6

0.00\,m

Reviewed time

t/T

0.00\,\%

Orange handles adjust the six sine amplitudes together; amplitude sliders provide keyboard access. Interval scale uses the harmonic half-period for harmonic motion and one second otherwise. Restore the reference zeros all six deviations only for compatible endpoints. The variation scan adds amplitude in one selected direction.

One particle in free motion, a uniform gravitational potential or an exact harmonic potential. Paths share fixed endpoints and six editable sine modes. The analytic reference is available only for compatible endpoints; at a conjugate time it is one member of a nonunique family. A projected six-mode gradient and a continuous Newton residual are separate diagnostics.

Start with the essentials

Focus question: Must the physical path have the smallest action?
One-sentence intuition: Zero first variation identifies motion; the second variation distinguishes minimum, flat direction and saddle.

Core mathematical model

One endpoint-fixed functional

S[q]=\int_0^T\left(\frac{m\dot q^2}{2}-V(q)\right)dt,\qquad q(0)=q_0,\ q(T)=q_T

Action integrates the Lagrangian, not total energy. Endpoints stay fixed when the interior path changes.

Stationarity recovers Newton motion

\delta S=-\int_0^T\left(m\ddot q+V^{\prime}(q)\right)\eta(t)dt,\qquad\eta(0)=\eta(T)=0

The force residual vanishes for a true path. The six-mode gradient tests only the editable subspace, while the plotted residual tests the continuous equation.

Editable paths preserve the endpoints

q(t)=q_b(t)+\sum_{n=1}^{6}a_n\sin\frac{n\pi t}{T}

The base is the analytic true path when compatible. At an impossible harmonic boundary it is a straight interpolation, explicitly labeled as having no physical solution.

Curvature decides minimum or saddle

\Delta S=\frac m4\sum_{n=1}^{6}a_n^2\left(\frac{n^2\pi^2}{T}-\omega^2T\right)

For a compatible harmonic reference the first variation is zero. A mode whose half-wavelength is too long lowers action; higher modes still raise it. Free motion and uniform gravity replace the frequency term by zero.

Conjugate times need compatible endpoints

\omega T=k\pi\quad\Longrightarrow\quad q_T=(-1)^kq_0

At this exact condition an endpoint pair may have infinitely many physical paths or none. A flat mode is not a uniquely selected path. The interval control uses the harmonic half-period as its base scale.

Audit numerical quadrature independently

\epsilon_Q=\left|S_{\rm Simpson}-S_{\rm closed}\right|

Closed-form quadratic action is the displayed value. Composite Simpson integration with 512 intervals audits the path integral; this discrepancy is not a complete error bound.

Common difficulties

Stationary is not always minimum

Typical misconceptionThe true path must beat every nearby path.

Better mental modelBeyond a harmonic half-period there are negative-curvature directions and a stationary saddle.

Flat is not unique

Typical misconceptionA zero slope and zero curvature pick one motion.

Better mental modelA compatible conjugate boundary has a family of true paths along a flat direction.

Impossible endpoints stay impossible

Typical misconceptionAny two endpoint positions admit a harmonic path.

Better mental modelAt a conjugate time incompatible endpoint values have no solution; a straight base is only an editable candidate.

Projection has a scope

Typical misconceptionSix small derivatives prove stationarity in every possible direction.

Better mental modelThe finite basis tests six directions. Inspect the continuous equation residual and the analytic reference separately.

Run the experiment

01
Restore a minimum
Choose the short interval and restore the reference.
What to observe: Slope and force residual approach zero; a first-mode perturbation increases action.
02
Find a saddle
Choose the stationary saddle, then compare modes one and three.
What to observe: One lowers action and the other raises it around the same stationary path.
03
Test a conjugate boundary
Compare the family and impossible-endpoint presets.
What to observe: A flat physical family differs from an endpoint pair with no classical solution.
04
Edit and audit
Drag the orange handle and compare gradient, residual and quadrature discrepancy.
What to observe: The endpoints stay fixed; the exact action and independent quadrature remain consistent.