Level 3 · Undergraduate core teaching kit · First- and second-year university
Lagrangian and Hamiltonian mechanics
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Fixed-endpoint stationary-action test
How does the classical oscillator path respond to positive and negative variations that share its endpoints?
Setup
Use the action-path laboratory. Keep endpoints and duration fixed, evaluate the classical path, then apply matched positive and negative sine variations at several amplitudes.
Predict first
- 1. Predict ΔS at zero variation.
- 2. Predict the relation between small +α and −α action changes near a stationary path.
| Variable | Role | Unit |
|---|---|---|
| Endpoint positions and duration | controlled | model position and s |
| Variation amplitude | independent | model position |
| Discretized action | dependent | model action |
| Action difference ΔS | diagnostic | model action |
Observation columns
Analyze
- 1. What evidence supports zero first-order change?
- 2. Why must every trial share endpoints?
- 3. Does stationary always mean minimum?
- 4. How could discretization imitate a nonstationary result?
Conclusion frame
Matched variations ±___ produced ΔS values ___ and ___ around the classical path; this supports/does not support stationarity because ___.
Instructor guide · 50–65 minutes
Teach the investigation, not the interface
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.
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
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.
Extension
Change duration through a conjugate-point threshold and investigate when the classical path ceases to be a local minimum.