Numerical drift in rotation

Rotation models can accumulate error just like linear models. A visually plausible spinning object may gradually gain or lose energy because the update method does not conserve the relevant quantity at the selected step. Drift is a time-series diagnostic relative to a stated system, force/torque model, integrator, and output horizon—not a verdict inferred from a smooth rotation.

Think like a programmer

Choose rotational invariants before rendering: angular momentum for torque-free fixed-axis dynamics, rotational kinetic energy for the appropriate conservative model, and any boundary/external-torque transfer. Record initial value, per-checkpoint value, absolute/relative residual, cumulative transfer, integrator, step, and physical time. Keep unwrapped orientation state separate from display wrapping so the seam cannot masquerade as drift.

Model checklist

Inputs: Angular state, inertia model, internal/external torque policy, boundary/friction/drive transfers, time step/integrator, output checkpoints, reference solution or refinement schedule, and tolerance.
State: Unwrapped orientation, angular velocity, torque, angular momentum, rotational kinetic energy, transfer ledger, drift/residual history, and display orientation.
Rule: Advance angular state under the declared integrator; calculate invariant/ledger values at common physical checkpoints; compare methods and steps without changing the physical experiment.
Output: Angular trace, invariant residual curves, transfer attribution, convergence comparison, and visual state derived from the trace.
Check: Torque-free fixed-inertia model keeps L = Iω within tolerance; no-friction/no-drive reference keeps compatible energy bounded; externally applied torque changes L by recorded angular impulse; reduced step improves a selected fixed-time error metric; wrapping display angle leaves physical invariant calculations unchanged.

\[L=I\omega,\qquad K_{rot}=\tfrac12I\omega^2,\qquad \Delta L=\int\tau_{ext}\,dt+r_L\]

Integrator energy drift

Run the same ideal oscillator with Euler and fourth-order Runge–Kutta steps, then compare an energy invariant.

Time step: 0.20 s

Euler state: x = 4.473, v = -5.522
RK4 state: x = 0.408, v = -0.913

Euler energy 25.2525; RK4 energy 0.5000; exact energy 0.5000.

Torque updates angular motion

Change torque and moment of inertia. The same torque produces less angular acceleration for a harder-to-rotate body.

Torque: 6 N·mMoment of inertia: 2 kg·m²

Torque 6 N·m; moment of inertia 2 kg·m²; angular acceleration 3.00 rad/s²; rotational energy at 3 rad/s 9.00 J.

Compare methods at fixed physical time, not equal step count. A smaller Δt gives more updates; plotting after the same number of updates changes the experiment length. Choose an error metric such as maximum relative energy drift, endpoint angular-momentum residual, or phase/orientation error against a reference. Long simulations may expose a small systematic bias that a short run hides, so include both horizon and checkpoint cadence in the run record.

External torque and friction alter the expected ledger. A changing angular momentum under a motor is physical if its angular impulse is recorded; a changing energy under friction is physical if transfer to a reservoir is modeled. The residual is what remains after those named effects, not the raw difference from the initial value.

Try this experiment

Prediction: A smaller time step generally reduces drift in an ideal model, while a nonzero recorded external torque changes angular momentum physically.

Run a torque-free fixed-inertia fixture at Δt and Δt/2 to the same final time. Compare L and rotational-energy residual curves. Repeat with a constant torque, integrate its angular impulse, and check the ledger rather than demanding constant L. Change only display wrapping and verify that invariant diagnostics remain identical.

Where this model breaks

Energy and angular momentum may change physically under external torque, friction, changing inertia, gyroscopic coupling, constraints, or boundary interaction. Define the modeled system before treating a changing value as numerical error. General 3D rigid bodies need vector/tensor angular momentum and orientation methods with their own numerical constraints; a scalar fixed-axis lesson cannot validate them.

Summary

Track rotational invariants and named transfers in time-stepped models. Compare integrators and steps at common physical checkpoints, preserve unwrapped state, and distinguish physical torque/friction changes from residual drift before trusting a rotating visual.

Glossary

Angular momentum: rotational analogue of linear momentum.
Drift: accumulated numerical error.
Torque-free: no external torque acts on the modeled system.
Angular impulse: time integral of external torque.
Checkpoint: named physical time at which diagnostics are compared.

Self-check

Which quantity is conserved without external torque in the fixed-axis model?
Why compare integrators at the same physical time?
When can a changing invariant be physical?
Why should display wrapping not enter invariant calculations?

Sources

E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration.
J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Numerical Drift in Rotation, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Rotational Motion, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Numerical Drift in Rotation as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Numerical Drift in Rotation into a test

Use rotational energy and angular momentum checks to identify error that accumulates in a time-stepped rotation model.

Name the inputs and units that the rotational motion model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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