# Instructor guide: Fixed-endpoint stationary-action test

Course: Lagrangian and Hamiltonian mechanics

Suggested time: 50–65 minutes

## Learning target

Learners interpret stationary action as cancellation of first-order fixed-endpoint variations rather than a force-free or universally minimum path.

## Prepare

- Review L=T−V and fixed endpoint variations.
- Sketch the classical and one trial path.
- Choose parameters inside the laboratory's local-minimum range.

## Facilitation moves

- Check endpoints before discussing action values.
- Pair +α and −α runs.
- Connect the global comparison to the local Euler–Lagrange equation.

## Misconception checks

- **Nature tests every path consciously.** The variational statement is a compact mathematical property of solutions, not a time-ordered search process.
- **Stationary action is always the smallest action.** Stationary points may be minima, maxima, or saddles depending on the system and interval.

## Accessibility and participation

- Use path labels and line styles as well as color.
- Provide the action ledger before graph interpretation.
- Allow symbolic explanation when graph drawing is difficult.

## Evidence of learning

- Matched fixed-endpoint variations
- Correct first-order cancellation language
- A stationarity-versus-minimum distinction

## Extension

Change duration through a conjugate-point threshold and investigate when the classical path ceases to be a local minimum.

## Evidence boundary

Assess the learner's reasoning only within the declared model and recorded observations. Do not upgrade a simulation result into a claim about an unmodeled physical system.