M047 · Energy averages

Virial Theorem Observatory

Follow bound and escaping trajectories in power and logarithmic potentials. Compare running kinetic and radial-force averages, retain the finite-time boundary term and audit the relation again from raw sensor readings.

Mean potential energy $\text{Ready}$

Selected time $\text{Ready}$

Particle radius $\text{Ready}$

Particle speed $\text{Ready}$

Kinetic energy $\text{Ready}$

Potential energy $\text{Ready}$

Twice mean kinetic energy $\text{Ready}$

Mean radial virial $\text{Ready}$

Finite-window boundary term $\text{Ready}$

Finite-window identity residual $\text{Ready}$

Characteristic clock $\text{Ready}$

Maximum energy error $\text{Ready}$

Maximum angular momentum error $\text{Ready}$

Endpoint refinement difference $\text{Ready}$

Raw-record identity residual $\text{Ready}$

Initial binding condition $\text{Ready}$

Acquired trajectory stations $\text{Ready}$

DENSITY / ORBITS / AVERAGES

When does the energy average settle?

Power exponent; zero selects logarithm

n

-1.00\,

Reference radius

r_0

1.00\,m

Potential energy scale

A

1.00\,J

Initial tangential speed over circular speed

v_{t,0}/v_c

0.80\,

Initial outward radial speed over circular speed

v_{r,0}/v_c

0.00\,

Observation window in characteristic clocks

T/t_0

35.00\,

Integration step over characteristic clock

\Delta t/t_0

0.0050

Selected time fraction

t/T

0.65\,

Sensor noise gain

\eta

0.00\,

One kilogram test particle in a prescribed attractive radial field. Power potentials use their homogeneous zero; the zero exponent selects a separate logarithmic potential with zero at the reference radius.

Energy averages use every integration step. The finite-window boundary term must be retained; bound motion is a condition, not evidence that a short observation has converged.

Velocity Verlet; no artificial softening or damping. Integration stops before a sampled radius leaves the declared domain. Endpoint refinement is a diagnostic, not a rigorous bound.

Acquire the full solved window. Raw estimates integrate measured position, velocity and force only; sensor noise gain scales millimetres, millimetres per second and millinewtons. No model exponent or potential enters the estimate.

Prescribed Newtonian radial field; numerical errors and approximation boundaries are explicit.

Start with the essentials

Focus question: When does the energy average settle?
One-sentence intuition: A finite boundary term is physical, and must not be mistaken for numerical error.

Core mathematical model

Attractive radial potentials

\begin{aligned}U(r)&=\frac A n\left(\frac r{r_0}\right)^n\quad(n\ne0)\\U_{\log}(r)&=A\ln\left(\frac r{r_0}\right)\end{aligned}

The logarithmic member is a separate potential; the power-potential energy zero is not shifted. The test mass is one kilogram.

Instantaneous virial identity

\frac{d(\mathbf r\cdot\mathbf p)}{dt}=2K-W,\qquad W=-\mathbf r\cdot\mathbf F

The outward radial coordinate and inward force determine the signs.

Finite-time audit

2\langle K\rangle_T-\langle W\rangle_T=\frac{(\mathbf r\cdot\mathbf p)(T)-(\mathbf r\cdot\mathbf p)(0)}{T}

A short interval can have a large physical boundary term while satisfying the identity accurately.

Power and logarithmic limits

\begin{aligned}W&=nU\quad(n\ne0)\\W_{\log}&=A\end{aligned}

Only when the boundary term vanishes does the familiar long-time energy relation follow.

Conserved quantities

\begin{aligned}E&=\frac12m|\mathbf v|^2+U(r)\\L_y&=m(zv_x-xv_z)\end{aligned}

The planar orbit lies in the horizontal plane. Energy and angular momentum errors are reported independently of the virial residual.

Common difficulties

Assumption boundary

Typical misconceptionAn ideal model automatically describes a real planet or relaxed star cluster.

Better mental modelThe source is prescribed, the particle has no back reaction, and omitted effects are stated.

Run the experiment

01
Compare potentials
Try Kepler ellipse, Harmonic well, Quartic confinement and Logarithmic potential.
What to observe: The force law and running averages change together.
02
Keep the boundary
Compare Short observation and a longer duration; scrub the selected time.
What to observe: The average difference follows the physical boundary term; its corrected residual diagnoses numerics.
03
Test the assumption
Increase the Kepler launch speed beyond escape and watch the boundary term.
What to observe: An escaping particle violates the bounded-motion premise; extending time need not repair it.
04
Acquire and refine
Acquire the solved trajectory, inspect raw estimates, then halve the integration step.
What to observe: Raw resampling and sensor noise are separate from the every-step numerical average.