M081 · Action with fixed endpoints

Principle of Stationary Action

Must the physical path have the smallest action? No; it only has to be stationary, meaning a small change to the path leaves the action unchanged to first order. For short trips that stationary path is a minimum. For long enough trips, past a so-called conjugate point, it can become a saddle instead. For a mass on a spring, that happens once the trip lasts longer than half an oscillation period.

Drag a trial path while its endpoints stay fixed. Compare the action, its first variation and the residual of the equation of motion for free motion, uniform gravity and harmonic motion. Then move from a minimum to a saddle, and test compatible and impossible endpoints at a conjugate time.

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 $\text{Pending}$

Change the setup and compare trajectories

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\,\%

The orange handles adjust all six sine amplitudes together, and the amplitude sliders give keyboard access. The interval is scaled by the harmonic half-period for harmonic motion and by one second otherwise. Restore reference sets all six deviations to zero only when the endpoints are compatible. The variation scan adds amplitude along one chosen direction.

A single particle moves freely, in a uniform gravitational potential or in an exact harmonic potential. All paths share the same fixed endpoints and differ only in six editable sine modes. An analytic reference exists only for compatible endpoints; at a conjugate time it is just one member of a family of solutions that is not unique. The gradient projected onto the six modes and the continuous Newton residual are separate checks.

Start with the essentials

Focus question: Must the physical path have the smallest action?
One-sentence intuition: A zero first variation picks out the actual motion. The second variation then tells whether that path is a minimum, has a flat direction or is a saddle.

Core mathematical model

Action with fixed endpoints

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

The action is the time integral of the Lagrangian, not of the total energy. When the path in between changes, the endpoints stay put.

A stationary action gives Newton's law

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

On a true path the force residual is zero. The gradient over the six modes tests only the directions you can edit; the plotted residual tests the full continuous equation.

Editable paths keep the endpoints fixed

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

When the endpoints are compatible, the base path is the analytic true path. When harmonic endpoints are impossible, the base is a straight line between them, and the case is marked 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)

Around a compatible harmonic reference, the first variation is zero. A mode with too long a half-wavelength lowers the action, while higher modes still raise it. For free motion and uniform gravity, the frequency term is zero.

Conjugate times need compatible endpoints

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

When this condition holds exactly, a pair of endpoints can have infinitely many physical paths or none. A flat mode does not single out one path. The interval control measures time in units of the harmonic half-period.

An independent check on the numerical integral

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

The displayed action comes from the closed-form quadratic expression. A composite Simpson integration with 512 intervals checks the path integral independently; the difference between them is not a complete error bound.

Common difficulties

Stationary does not always mean minimum

Typical misconceptionThe true path must have less action than every nearby path.

Better mental modelOnce the interval is longer than half a harmonic period, some directions have negative curvature and the stationary path is a saddle.

A flat direction does not mean a unique path

Typical misconceptionIf both the slope and the curvature are zero, they pick out one motion.

Better mental modelWith compatible endpoints at a conjugate time, a whole family of true paths lies along the flat direction.

Impossible endpoints stay impossible

Typical misconceptionAny two endpoint positions can be joined by a harmonic path.

Better mental modelAt a conjugate time, incompatible endpoints have no solution. The straight base line is only a candidate you can edit.

Six modes test only six directions

Typical misconceptionSix small derivatives prove the path is stationary in every possible direction.

Better mental modelThe finite basis tests only six directions. Check the residual of the continuous equation and the analytic reference as well.

Run the experiment

01
Return to a minimum
Choose Short-interval minimum, then press Restore reference.
What to observe: The slope and the force residual approach zero, and pushing the path along the first mode raises the action.
02
Find a saddle
Choose Stationary saddle, then compare modes one and three.
What to observe: Around the same stationary path, one mode lowers the action and the other raises it.
03
Test endpoints at a conjugate time
Compare the Conjugate-time family and Impossible endpoints presets.
What to observe: A family of physical paths along a flat direction is a different case from endpoints that have no classical solution at all.
04
Edit the path and check the numbers
Drag the orange handle and compare the gradient, the residual and the quadrature difference.
What to observe: The endpoints stay fixed, and the exact action agrees with the independent numerical integral.