Test orbital invariants against an independent exact trajectory
A numerical orbit can preserve one invariant while accumulating trajectory error. Use the normalized central field μ = 1 and initial position (1, 0), velocity (0, 1). The exact orbit is the unit circle with period 2π, energy per mass −0.5 and angular momentum per mass 1.
Learning outcome: Compare Euler and Verlet on two grids, saving invariant extrema and endpoint position/velocity errors with the exact reference.
Starting recap
Integrate mass moments about named axes, derive torque and angular-momentum constraints, and test orbital energy, angular and numerical consistency against independent evidence.
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The exact endpoint after one period is position (1, 0) and velocity (0, 1). Any nonzero discrepancy in this stipulated noiseless benchmark is numerical, not an observation error. Halving the step changes the approximation, not the target reference.
Open the experimentPredict
If Verlet preserves angular momentum to rounding in this benchmark, that alone establishes:
Experiment
Compare Euler and Verlet on two grids, saving invariant extrema and endpoint position/velocity errors with the exact reference.
Compare orbital invariants with trajectory error
Worked teaching model · synthetic values · Normalized mu=1, initial r=(1,0),v=(0,1),exact circular period2pi. Compare forward Euler and velocity Verlet over that fixed interval. Point source and bounded numerical benchmark, not real orbital prediction.
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One exact-period numerical benchmark
Spatial axes use the same pixels per physical unit, preserving geometry. 1: computed trajectory (solid).2: exact circular reference (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for One exact-period numerical benchmark
computed trajectory
| east position (m) | north position (m) |
|---|---|
| 1 | 0 |
| 0.987663 | 0.1570796 |
| 0.9509619 | 0.3102844 |
| 0.8908181 | 0.4558401 |
| 0.8087386 | 0.5901711 |
| 0.7067747 | 0.7099916 |
| 0.587466 | 0.8123884 |
| 0.4537722 | 0.8948925 |
| 0.3089943 | 0.9555374 |
| 0.1566887 | 0.9929038 |
| 0.0005766681 | 1.006149 |
| -0.1555494 | 0.9950206 |
| -0.3079181 | 0.9598573 |
| -0.4528695 | 0.9015723 |
| -0.5869406 | 0.821627 |
| -0.7069445 | 0.7219896 |
| -0.8100416 | 0.6050857 |
| -0.893802 | 0.4737377 |
| -0.956258 | 0.3310977 |
| -0.9959457 | 0.1805744 |
| -1.011936 | 0.02575452 |
| -1.003855 | -0.129678 |
| -0.9718885 | -0.2820251 |
| -0.9167838 | -0.4276577 |
| -0.8398291 | -0.5630979 |
| -0.7428295 | -0.6850982 |
| -0.628068 | -0.7907169 |
| -0.4982564 | -0.8773882 |
| -0.3564768 | -0.9429846 |
| -0.2061121 | -0.9858711 |
| -0.05076987 | -1.004949 |
| 0.1058019 | -0.999689 |
| 0.259804 | -0.9701485 |
| 0.4074783 | -0.916979 |
| 0.5452015 | -0.8414159 |
| 0.6695771 | -0.7452533 |
| 0.7775238 | -0.6308048 |
| 0.8663563 | -0.5008491 |
| 0.9338579 | -0.3585618 |
| 0.9783401 | -0.207436 |
| 0.9986893 | -0.05119336 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
exact circular reference
| east position (m) | north position (m) |
|---|---|
| 1 | 0 |
| 0.9969173 | 0.0784591 |
| 0.9876883 | 0.1564345 |
| 0.9723699 | 0.2334454 |
| 0.9510565 | 0.309017 |
| 0.9238795 | 0.3826834 |
| 0.8910065 | 0.4539905 |
| 0.8526402 | 0.5224986 |
| 0.809017 | 0.5877853 |
| 0.760406 | 0.649448 |
| 0.7071068 | 0.7071068 |
| 0.649448 | 0.760406 |
| 0.5877853 | 0.809017 |
| 0.5224986 | 0.8526402 |
| 0.4539905 | 0.8910065 |
| 0.3826834 | 0.9238795 |
| 0.309017 | 0.9510565 |
| 0.2334454 | 0.9723699 |
| 0.1564345 | 0.9876883 |
| 0.0784591 | 0.9969173 |
| 6.12323e-17 | 1 |
| -0.0784591 | 0.9969173 |
| -0.1564345 | 0.9876883 |
| -0.2334454 | 0.9723699 |
| -0.309017 | 0.9510565 |
| -0.3826834 | 0.9238795 |
| -0.4539905 | 0.8910065 |
| -0.5224986 | 0.8526402 |
| -0.5877853 | 0.809017 |
| -0.649448 | 0.760406 |
| -0.7071068 | 0.7071068 |
| -0.760406 | 0.649448 |
| -0.809017 | 0.5877853 |
| -0.8526402 | 0.5224986 |
| -0.8910065 | 0.4539905 |
| -0.9238795 | 0.3826834 |
| -0.9510565 | 0.309017 |
| -0.9723699 | 0.2334454 |
| -0.9876883 | 0.1564345 |
| -0.9969173 | 0.0784591 |
| -1 | 1.22465e-16 |
| -0.9969173 | -0.0784591 |
| -0.9876883 | -0.1564345 |
| -0.9723699 | -0.2334454 |
| -0.9510565 | -0.309017 |
| -0.9238795 | -0.3826834 |
| -0.8910065 | -0.4539905 |
| -0.8526402 | -0.5224986 |
| -0.809017 | -0.5877853 |
| -0.760406 | -0.649448 |
| -0.7071068 | -0.7071068 |
| -0.649448 | -0.760406 |
| -0.5877853 | -0.809017 |
| -0.5224986 | -0.8526402 |
| -0.4539905 | -0.8910065 |
| -0.3826834 | -0.9238795 |
| -0.309017 | -0.9510565 |
| -0.2334454 | -0.9723699 |
| -0.1564345 | -0.9876883 |
| -0.0784591 | -0.9969173 |
| -1.83697e-16 | -1 |
| 0.0784591 | -0.9969173 |
| 0.1564345 | -0.9876883 |
| 0.2334454 | -0.9723699 |
| 0.309017 | -0.9510565 |
| 0.3826834 | -0.9238795 |
| 0.4539905 | -0.8910065 |
| 0.5224986 | -0.8526402 |
| 0.5877853 | -0.809017 |
| 0.649448 | -0.760406 |
| 0.7071068 | -0.7071068 |
| 0.760406 | -0.649448 |
| 0.809017 | -0.5877853 |
| 0.8526402 | -0.5224986 |
| 0.8910065 | -0.4539905 |
| 0.9238795 | -0.3826834 |
| 0.9510565 | -0.309017 |
| 0.9723699 | -0.2334454 |
| 0.9876883 | -0.1564345 |
| 0.9969173 | -0.0784591 |
| 1 | -2.44929e-16 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Step size | 0.1570796 | s |
| Maximum sampled specific energy error | 7.33904e-5 | m²/s² |
| Maximum sampled specific angular momentum error | 3.33067e-16 | m²/s |
| Endpoint position error | 0.05121013 | m |
| Endpoint velocity error | 0.05112682 | m/s |
| Final specific energy | -0.5 | m²/s² |
Conserved angular momentum or small energy drift does not alone establish the correct orbital phase.
The exact reference returns to its initial state at2pi. Forward Euler uses old velocity and acceleration; velocity Verlet uses a position drift with half acceleration and averages old/new accelerations for velocity. The central-force Verlet steps preserve angular momentum up to rounding, but finite-step trajectory error remains. Compare two grids and both invariants with the independent exact endpoint.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Conserved angular momentum or small energy drift does not alone establish the correct orbital phase.
The exact reference returns to its initial state at2pi. Forward Euler uses old velocity and acceleration; velocity Verlet uses a position drift with half acceleration and averages old/new accelerations for velocity. The central-force Verlet steps preserve angular momentum up to rounding, but finite-step trajectory error remains. Compare two grids and both invariants with the independent exact endpoint.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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Public practice checks a specific claim. It does not establish independent mastery.
How can a method preserve angular momentum yet return to the wrong point after one exact period?
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