PYTHON · NUMPY

Energy, and why integrators drift

A frictionless system conserves energy forever — but a careless simulator invents or leaks it. See why the update rule decides, meet structure-preserving integration, and understand the failure mode that haunts every learned dynamics model.

01Challenge

Take the pendulum from the last lesson, remove all friction, and let it swing. Physics is emphatic: with no dissipation, total energy T + U is constant for all time. Now simulate it two ways from the identical start. Explicit (forward) Euler moves the position using the OLD velocity; symplectic (semi-implicit) Euler updates the velocity first, then moves with the NEW velocity — a one-line difference. The Bench runs both on the same ferromotion dynamics and plots their energy over time. Watch the red curve (explicit Euler) climb away from the true energy — phantom energy, the swing amplifying until it flies apart — while the green curve (symplectic) stays pinned in a razor-thin band. Same physics, same forces; only the integrator differs.

Predict firstYou simulate a FRICTIONLESS pendulum with a simple explicit-Euler integrator and let it swing for a long time. What happens to its total energy?

02Model

Energy conservation is a structural property of the true dynamics — a consequence of the Hamiltonian being constant along trajectories. A generic integrator does not know this and violates it a little every step; those errors accumulate. Explicit Euler is biased to add energy (it consistently overshoots), so an undamped oscillator spirals outward and eventually blows up. Symplectic (variational) integrators are built to preserve the geometric structure of Hamilton's equations, so their energy error oscillates within a bounded band that never grows, even over millions of steps. This is not a rounding detail — it is THE failure mode of learned dynamics models. A neural network trained to predict the next state, rolled out over a long horizon, drifts exactly like explicit Euler: it leaks or injects energy and the rollout diverges from physical reality. That is why later modules put conservation INTO the model — Hamiltonian and Lagrangian nets that conserve energy by construction, and Variational Integrator Networks that borrow the symplectic update you are watching here.

03Guided practice

Step A (watch the divergence): both integrators start at the same energy E₀ (the dashed line). Within a few seconds the red explicit-Euler energy visibly climbs while the green symplectic line hugs E₀. Step B (read the drift): the readout shows explicit Euler's drift growing into whole joules; the symplectic drift stays in the hundredths. Step C (let it run): the longer it runs, the wider the gap — explicit error grows without bound, symplectic error does not. Step D (the lesson): the forces were identical; ferromotion computed the same q̈ for both. The ONLY difference was the order of the position and velocity updates. Structure-preserving integration is why a physical rollout stays physical.

04Feedback

PASS WHEN PASS once explicit Euler has visibly injected energy while the symplectic integrator held the true energy — the signature of structure-preserving integration.

If your run fails, check:

Let it run for a few seconds — the red explicit-Euler curve climbs while the green symplectic curve stays flat.
Press Reset to restart both integrators from the same energy and watch them diverge again.

05Retrieve & space

A frictionless pendulum must conserve energy — why does explicit Euler violate that, and in which direction?
What is the single-line difference between explicit and symplectic Euler, and why does it matter over long horizons?
How does this failure mode show up when a neural network predicts dynamics and is rolled out for many steps?

06Mastery & project

You saw that the update rule, not just the forces, decides whether a simulation stays physical — and that structure-preserving integration bounds the energy error forever. This is the exact drift that sinks naive learned dynamics models, and the reason the encoded-physics module builds conservation into the network itself.

Directly motivates two later methods: Variational Integrator Networks (learn dynamics with a symplectic integrator in the loop) and Hamiltonian/Lagrangian networks (conserve energy by construction). Keep this energy plot in mind when a learned model's rollout blows up.

GO FURTHER · THE LIVE PLAYGROUND

ferromotion lgvi: Lie-group variational integrator ↗ — the structure-preserving integrator ferromotion ships for rotations — the grown-up version of the symplectic Euler here.

← The equation of motion Solve an equation with no data →

PYTHON · NUMPY

Energy, and why integrators drift

Live ferromotion-wasm: the same frictionless pendulum under explicit vs symplectic Euler. Watch explicit Euler manufacture energy while the structure-preserving integrator conserves it.

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