# Learner lab record: Fixed-endpoint stationary-action test

Course: Lagrangian and Hamiltonian mechanics

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

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.

## Variables

| 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 |

## Predict before changing controls

1. Predict ΔS at zero variation.

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2. Predict the relation between small +α and −α action changes near a stationary path.

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## Observation table

| duration | variation α | S trial | S classical | ΔS | endpoint match? |
| --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |

## Analyze

1. What evidence supports zero first-order change?

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2. Why must every trial share endpoints?

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3. Does stationary always mean minimum?

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4. How could discretization imitate a nonstationary result?

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## Evidence-bounded conclusion

Matched variations ±___ produced ΔS values ___ and ___ around the classical path; this supports/does not support stationarity because ___.

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