M084 · Hamiltonian / numerical geometry

Hamiltonian Phase Space & Integrators

Run Euler, RK4, symplectic Euler and Verlet side by side. Compare long-time phase or orbital fidelity, energy drift, local canonical-map geometry and reversibility; halve the step and confront a rapidly changing Kepler periapsis.

Reviewed scaled time $\text{Pending}$

Actual constant step $\text{Pending}$

Total steps per method $\text{Pending}$

Selected method $\text{Pending}$

Current relative energy error $\text{Pending}$

Maximum absolute energy error $\text{Pending}$

Current state error $\text{Pending}$

Last retained state error $\text{Pending}$

Local full-map determinant $\text{Pending}$

Local symplectic defect $\text{Pending}$

One-step reversal error $\text{Pending}$

Reference construction $\text{Pending}$

Reference refinement disagreement $\text{Pending}$

Selected method record $\text{Pending}$

Retained record end time $\text{Pending}$

Canonical dimension $\text{Pending}$

OBSERVE / VARY / VERIFY

Can a small energy error hide a wrong trajectory?

System: oscillator / pendulum / Kepler

\mathcal M

0.00\,

Initial coordinate

q_0

1.00\,

Initial momentum

p_0

0.00\,

Initial Kepler eccentricity

e

0.40\,

Requested step fraction

h/T_0

0.02\,

Nominal cycles

T/T_0

20.00\,

Method: Euler / RK4 / symplectic Euler / Verlet

\mathcal I

3.00\,

Canonical patch radius

\rho

0.03\,

Reviewed time

t/T

0.00\,\%

Drag the orange initial-state ring; the matching coordinate and momentum sliders provide keyboard access. In Kepler mode the ring edits eccentricity, while oscillator controls are inactive. All records are precomputed; playback reviews retained samples. In the lower-right view centers are removed. The Kepler patch is only a projection; the determinant and symplectic defect use the full four-dimensional canonical map. Diverged or singular records stop and remain labeled.

Dimensionless unit-mass conservative systems: harmonic oscillator, ideal pendulum and planar point-mass Kepler orbit. All four methods use the same constant step; nominal cycles use the harmonic or Kepler reference period, not the finite-amplitude pendulum period. Oscillator and Kepler references are analytic; pendulum reference uses RK4 at two finer steps. A refinement disagreement is an estimate, not a rigorous error bound. No adaptive stepping, damping or physical collision model.

Start with the essentials

Focus question: Can a small energy error hide a wrong trajectory?
One-sentence intuition: Accuracy, energy behavior, symplecticity and reversibility are related but distinct measurements.

Core mathematical model

The canonical flow

\dot{\mathbf q}=\mathbf p,\qquad\dot{\mathbf p}=-\nabla V,\qquad H=\frac{|\mathbf p|^2}{2}+V

Unit mass and scaled units make the three conservative systems directly comparable. Harmonic frequency and Kepler gravity parameter are one.

A kick followed by a drift

\mathbf p_{j+1}=\mathbf p_j-h\nabla V(\mathbf q_j),\qquad\mathbf q_{j+1}=\mathbf q_j+h\mathbf p_{j+1}

Symplectic Euler is first order. The use of updated momentum in the drift distinguishes it from explicit Euler.

Symmetric Verlet steps

\mathbf p_{j+1/2}=\mathbf p_j-\frac h2\nabla V(\mathbf q_j),\quad\mathbf q_{j+1}=\mathbf q_j+h\mathbf p_{j+1/2},\quad\mathbf p_{j+1}=\mathbf p_{j+1/2}-\frac h2\nabla V(\mathbf q_{j+1})

Two half-kicks make a second-order reversible map. This does not make the true Hamiltonian exactly conserved or an unresolved periapsis accurate.

Test the full symplectic geometry

D\Phi_h^{\mathsf T}J D\Phi_h=J,\qquad J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}

The readout reports the largest entrywise defect using central differences. A unit determinant alone is weaker in four dimensions. The Kepler patch is a two-dimensional projection, never a full-volume measurement.

Energy and state are different errors

\epsilon_H=\frac{H_j-H_0}{\max(|H_0|,10^{-6})},\qquad\epsilon_z=\|\mathbf z_j-\mathbf z_{\rm ref}(t_j)\|_2

All steps contribute to the maximum energy error; plots retain at most about 600 samples per method. A stable energy trace can hide accumulating phase error. Stopped records have no predicted continuation.

Read the logarithmic axes

y_H=\operatorname{sgn}(\epsilon_H)\log_{10}\left(1+\frac{|\epsilon_H|}{10^{-8}}\right),\qquad y_z=\log_{10}\left(1+\frac{\epsilon_z}{10^{-8}}\right)

A common logarithmic transform shows tiny and large errors together while retaining the sign of energy drift. Nominal cycles use a reference period of two pi in scaled time. Pendulum references compare 32 and 64 RK4 subdivisions of each coarse step.

Common difficulties

Energy is not the whole trajectory

Typical misconceptionSmall energy error guarantees a correct orbit.

Better mental modelPhase error can accumulate at nearly conserved energy. Compare the reference state at matching times.

Symplectic does not mean exact energy

Typical misconceptionA symplectic method conserves the exact Hamiltonian every step.

Better mental modelA resolved fixed-step method often has bounded oscillatory energy error, while the true energy is not exact.

Resolve periapsis

Typical misconceptionPreserving geometry cures a large time step.

Better mental modelThe fast near-center encounter still needs a short step. Halve the step and compare orbital error, not just determinant.

Projection is not canonical volume

Typical misconceptionThe area of a projected Kepler patch proves symplecticity.

Better mental modelKepler phase space has four dimensions. The full matrix test supplies the relevant local evidence.

Run the experiment

01
Compare all methods
Review a full harmonic record and change the selected method.
What to observe: All colors keep the same step; energy growth, drift and bounded oscillation differ.
02
Halve the step
Use the long RK4 record, then halve the fixed step.
What to observe: Both state error and energy drift respond to resolution; high local order is not the only criterion.
03
Leave the small-angle limit
Compare large-amplitude libration and rotating pendulum presets.
What to observe: The numerical reference is explicitly refined rather than replaced by a harmonic approximation.
04
Challenge a Kepler periapsis
Compare resolved and unresolved periapsis; inspect the full-map defect and halve the step.
What to observe: A symplectic map can preserve its geometry while producing a poor trajectory.