SCIENCE · INVESTIGATE AND EXPLAIN

Use an energy plot as one diagnostic, not a full certificate

For the supplied unit-mass,unit-stiffness harmonic model, x(0) = 1 and v(0) = 0 give exact x=cos t,v=−sin t and constant energy(x²+v²)/2=.5 J. The benchmark runs to10 s with a bounded step count. This exact solution is used as an independent numerical reference, not a measured oscillator record.

Learning outcome: Compare all three updates on two grids; retain energy extrema and endpoint phase-space error, not just one reassuring scalar value.

Starting recap

Use vector dot products for work, gradients for conservative force, definite integrals for accumulation and explicit energy/frame boundaries. Check accessible states before assigning trajectories.

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Try

For forward Euler with N = 10,h = 1, energy at t = 10 is(.5)(2¹⁰)=512 J, although the exact energy is.5 J. This discrepancy is entirely numerical.

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Predict

Forward Euler energy growth in this supplied undriven oscillator is:

Your prediction and reason

Experiment

Compare all three updates on two grids; retain energy extrema and endpoint phase-space error, not just one reassuring scalar value.

Use energy diagnostics without equating them to trajectory accuracy

Worked teaching model · synthetic values · Supplied harmonic model m1,k1,x0=1,v0=0, exact x=cos t,v=−sin t and energy .5 J. Finite10 s benchmark; step<=1 s. Compare endpoint and maximum sampled energy error plus phase/trajectory error. This is a numerical diagnostic, not a physical excitation experiment.

Steps over10 s: 20 steps Method0 forward Euler,1 velocity-first,2 Verlet: 0 method

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Energy diagnostic over the finite benchmark

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: numerical energy (solid).2: exact constant energy (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Energy diagnostic over the finite benchmark

numerical energy

time (s)	mechanical energy (J)
0	0.5
0.5	0.625
1	0.78125
1.5	0.9765625
2	1.220703
2.5	1.525879
3	1.907349
3.5	2.384186
4	2.980232
4.5	3.72529
5	4.656613
5.5	5.820766
6	7.275958
6.5	9.094947
7	11.36868
7.5	14.21085
8	17.76357
8.5	22.20446
9	27.75558
9.5	34.69447
10	43.36809

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 constant energy

time (s)	mechanical energy (J)
0	0.5
10	0.5

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.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Step size	0.5	s
Numerical final position	-9.206092	m
Numerical final velocity	-1.408562	m/s
Endpoint energy	43.36809	J
Endpoint signed energy error	42.86809	J
Maximum sampled absolute energy error	42.86809	J
Endpoint phase-space distance from exact	8.591834	scaled SI state difference

Energy behavior is an important diagnostic but does not alone certify trajectory phase or accuracy.

Forward Euler multiplies harmonic energy by1+h² per step. Velocity-first is symplectic here and velocity Verlet is time symmetric; on this finite grid their energy errors need not grow like forward Euler, but endpoint phase error remains. Compare both energy and the independent exact state at the same time. No physical source supplies the numerical energy growth.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Energy behavior is an important diagnostic but does not alone certify trajectory phase or accuracy.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Use an energy plot as one diagnostic, not a full certificate

Forward Euler updates x with old v and v with old x. Expanding xnew²+vnew² shows that energy is multiplied by1+h² every step. Thus energy grows numerically even though the physical equation has no drive. The growth is an algorithm artifact, not an energy source that the model discovered.

Velocity-first Euler updates v before x. Velocity Verlet uses xnew=x+h v−h²x/2 and vnew=v−h(x+xnew)/2. These methods have different long-time structure, and on the supplied h≤1 grid their energy errors do not behave like forward Euler’s systematic growth. That observation is conditional on this equation, method and grid.

Compare maximum sampled energy error and endpoint energy with the independent exact position and velocity at t = 10. A method can have modest or oscillatory energy error yet still accumulate phase error, so energy alone cannot certify a correct trajectory. Conversely, an endpoint error near zero can hide errors earlier in the interval.

Changed-case check

Predict before reading the comparison. For forward Euler with N = 10,h = 1, energy at t = 10 is(.5)(2¹⁰)=512 J, although the exact energy is.5 J. This discrepancy is entirely numerical.

Retained investigation

Compare all three updates on two grids; retain energy extrema and endpoint phase-space error, not just one reassuring scalar value. How can a small energy error coexist with a wrong oscillation phase?

Source and boundary

Primary conceptual reference checked30 October2026: OpenStax University Physics Volume1. All functions, numerical packets and teaching prose are original. Bounded supplied models are not measured experiments, apparatus instructions, independent practical skill or institutional credit. Numerical algorithm properties are derived for the disclosed model; do not treat them as universal solver guarantees.

The velocity-first Euler and kick-drift-kick velocity Verlet updates are also documented in Andrew Cumming, PHYS 512: Symplectic integrators, checked 30 October 2026. This lesson derives and tests their behavior for its own finite harmonic benchmark rather than borrowing the source’s orbit example.

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How can a small energy error coexist with a wrong oscillation phase?

Your explanation and limitations

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