# Instructor guide: Oscillator convergence and energy-drift study

Course: Calculus, vectors, and differential equations

Suggested time: 50–65 minutes

## Learning target

Learners separate physical parameters from numerical controls and justify convergence using trajectory and invariant diagnostics.

## Prepare

- Review ω=√(k/m) and oscillator energy.
- Define convergence before opening the lab.
- Prepare one stable-looking but inaccurate run.

## Facilitation moves

- Ask which equation stayed fixed across the comparison.
- Require at least three step sizes.
- Compare phase error with energy error rather than accepting one plot.

## Misconception checks

- **A smooth curve must be accurate.** Interpolation can look smooth while phase and energy drift far from the differential-equation solution.
- **More steps always repair an unstable model.** Smaller steps address discretization; they do not repair an incorrect equation, boundary condition, or unit conversion.

## Accessibility and participation

- Pair plots with numeric error summaries.
- Describe phase-space motion without requiring color distinction.
- Provide a three-run table and calculator support.

## Evidence of learning

- A three-step convergence table
- A physical-versus-numerical variable distinction
- A justified stability recommendation

## Extension

Compare symplectic Euler with forward Euler using identical initial conditions and computational cost.

## 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.