M047 · Energy averages

Virial Theorem Observatory

When does the virial theorem actually hold? Only for bound motion, and only after a long time. Over any finite stretch, twice the average kinetic energy differs from the average radial-force term by a boundary term: the change in position times momentum, divided by the elapsed time. That gap is real physics, not numerical error. A bound orbit keeps it finite, so it fades as time grows; an escaping one need not.

Follow bound and escaping orbits in power-law and logarithmic potentials. Compare the running averages of kinetic energy and of the radial-force term, keep the boundary term that a finite time window leaves, and check 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 $\text{Ready}$

Set the density, then follow orbits and their averages

When does the virial theorem actually hold?

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\,

A one-kilogram test particle moves in a fixed attractive radial field. Power-law potentials have no added constant, so each is a pure power of the radius. An exponent of zero switches to a logarithmic potential, which is zero at the reference radius.

The energy averages include every integration step. Over a finite time window the boundary term must be kept. Bound motion is needed for the averages to converge, but it does not show that a short observation already has.

The motion is integrated with velocity Verlet, with no artificial softening or damping. Integration stops before any sampled radius leaves the allowed range. The endpoint refinement difference is a rough accuracy check, not a rigorous error bound.

An acquisition records the whole solved time window. The raw estimates integrate only the measured position, velocity and force. Sensor noise is the noise gain times one millimetre, one millimetre per second and one millinewton. The estimate never uses the model's exponent or potential.

The Newtonian radial field is prescribed in advance. Numerical errors are shown, and the limits of each approximation are stated.

Start with the essentials

Focus question: When does the virial theorem actually hold?
One-sentence intuition: Over a finite time, the boundary term is real physics, not 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 potential is a separate choice, and the energy zero of the power-law potentials is not shifted to match it. The test mass is one kilogram.

The virial identity at each instant

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

The signs follow from a radius that points outward and a force that points inward.

The identity averaged over a finite time

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

Over a short interval the physical boundary term can be large, and the identity still holds accurately.

The virial term for each potential

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

The familiar long-time energy relation follows only when the boundary term goes to zero.

Conserved quantities

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

The orbit lies in the horizontal plane. Errors in energy and angular momentum are reported separately from the virial residual.

Common difficulties

What the model assumes

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

Better mental modelThe source is fixed in advance, the test particle does not pull back on it, and the effects left out are listed.

Run the experiment

01
Compare potentials
Try Kepler ellipse, Harmonic well, Quartic confinement and Logarithmic potential.
What to observe: When the force law changes, the running averages change with it.
02
Keep the boundary term
Compare Short observation with a longer duration, and drag the selected time.
What to observe: The gap between the averages tracks the physical boundary term. What remains after subtracting it measures numerical error.
03
Test the assumption
Raise the Kepler launch speed above escape speed and watch the boundary term.
What to observe: An escaping particle breaks the assumption of bounded motion, and observing for longer need not restore the relation.
04
Take readings, then refine the step
Acquire the computed trajectory, look at the raw estimates, then halve the integration step.
What to observe: Resampling and sensor noise affect the raw estimate, separately from the numerical average taken at every step.