Dynamic Virial Theorem

• Dynamic Virial Theorem is a time-dependent identity linking the second derivative of a system’s moment of inertia with its kinetic energy and force virials.
• It extends across various fields by adapting the virial balance to homogeneous power-law interactions, geometric frameworks, and dissipative as well as nonequilibrium systems.
• The theorem provides practical insights into energy balances, density of states, and transient dynamics, informing studies from quantum gases to gravitational models.

The dynamic virial theorem is the time-dependent identity that relates the second derivative of a system’s moment of inertia to its kinetic energy and the virial of the applied forces. In its standard particle form, for $I(t)=\sum_{i=1}^N m_i r_i^2$, it reads

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$

and its steady or long-time-averaged limit yields the familiar virial balance when the inertial term vanishes. Contemporary treatments place this identity in a broader setting: homogeneous power-law interactions, general steady states beyond canonical and microcanonical ensembles, geometric mechanics on manifolds and Lie algebroids, dissipative and open quantum systems, nonequilibrium quantum gases governed by Tan’s contact, continuum and micropolar media, and modified gravitational theories (Davis, 30 Jul 2025, Cariñena et al., 2014, Peng, 2022).

1. Classical identity and steady-state limit

For an $N$-particle system with masses $m_i$, positions $\mathbf r_i(t)$, momenta $\mathbf p_i=m_i\mathbf v_i$, and forces $\mathbf F_i$, the standard constructions are the scalar moment of inertia

$I(t)=\sum_{i=1}^N m_i r_i^2$

and the Clausius virial

$G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$

They satisfy $dI/dt=2G$, hence

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$0

with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$1 the instantaneous kinetic energy. For conservative forces $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$2, the potential form becomes

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$3

The time-averaged theorem follows when $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$4 remains bounded, or more generally when the long-time average of the total derivative vanishes; then

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$5

These statements appear in particle, continuum, and stochastic formulations with the same kinematic core (LeBohec, 2022).

For homogeneous potentials, Euler-type scaling converts the force virial into a potential-energy relation. If $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$6, then $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$7, so the averaged theorem becomes

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$8

For inverse-power attraction with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$9, one recovers $N$0, consistent with $N$1 and $N$2. A persistent source of confusion is the identification of the virial theorem with this averaged relation alone. The dynamic virial theorem is the instantaneous identity; the steady-state form requires the additional condition $N$3, and deviations from virial balance during collapse, expansion, or other transients are measured precisely by the inertial term $N$4 (Davis, 30 Jul 2025).

2. Homogeneous power-law interactions, density of states, and steady states beyond Gibbs ensembles

For pairwise power-law interactions,

$N$5

homogeneity implies

$N$6

Since $N$7, one has

$N$8

and therefore in the steady-state limit

$N$9

With the notation $m_i$0, this is also written $m_i$1. The relation holds in any spatial dimension $m_i$2 provided the interaction is a pure power and sufficiently regular for the derivatives to exist.

A notable development is the exact computation of the configurational density of states

$m_i$3

for homogeneous pair potentials. In three dimensions, using Rugh’s geometrical framework with the vector field $m_i$4, one obtains

$m_i$5

Combined with the quadratic kinetic density of states

$m_i$6

the total density of states follows by convolution,

$m_i$7

which implies

$m_i$8

Thus the microcanonical heat capacity is constant for this class of systems. For $m_i$9, the formula gives

$\mathbf r_i(t)$0

while canonical stability requires $\mathbf r_i(t)$1, which is ensured for any positive $\mathbf r_i(t)$2 and for negative $\mathbf r_i(t)$3 satisfying $\mathbf r_i(t)$4.

The same paper shows that the virial theorem can be recovered from the configurational density of states alone, without assuming canonical or microcanonical Gibbs measures. If the steady state has the form

$\mathbf r_i(t)$5

then

$\mathbf r_i(t)$6

and the conjugate variables theorem yields, after choosing $\mathbf r_i(t)$7,

$\mathbf r_i(t)$8

This extends the steady-state virial theorem to any stationary distribution that depends on phase-space variables through the conserved energy alone. By contrast, for non-homogeneous interactions such as Lennard–Jones $\mathbf r_i(t)$9, no single degree $\mathbf p_i=m_i\mathbf v_i$0 exists; the correct relation is

$\mathbf p_i=m_i\mathbf v_i$1

not $\mathbf p_i=m_i\mathbf v_i$2 with one exponent (Davis, 30 Jul 2025).

3. Geometric formulations on manifolds, in quasi-coordinates, and on Lie algebroids

In geometric mechanics, the dynamic virial theorem is formulated intrinsically for mechanical Lagrangians on a Riemannian configuration manifold $\mathbf p_i=m_i\mathbf v_i$3. With metric $\mathbf p_i=m_i\mathbf v_i$4, kinetic energy $\mathbf p_i=m_i\mathbf v_i$5, and Lagrangian $\mathbf p_i=m_i\mathbf v_i$6, a vector field $\mathbf p_i=m_i\mathbf v_i$7 defines the linear virial function

$\mathbf p_i=m_i\mathbf v_i$8

Along a solution $\mathbf p_i=m_i\mathbf v_i$9 of $\mathbf F_i$0, differentiation gives

$\mathbf F_i$1

Under periodicity or boundedness hypotheses ensuring that the average of a total derivative vanishes, the averaged virial relation becomes

$\mathbf F_i$2

Special cases follow from the symmetry class of $\mathbf F_i$3: for Killing fields $\mathbf F_i$4, one gets $\mathbf F_i$5; for homothetic fields $\mathbf F_i$6 with constant $\mathbf F_i$7, $\mathbf F_i$8; for conformal Killing fields $\mathbf F_i$9, the kinetic term is weighted locally by $I(t)=\sum_{i=1}^N m_i r_i^2$0. The Euclidean dilation field $I(t)=\sum_{i=1}^N m_i r_i^2$1 recovers

$I(t)=\sum_{i=1}^N m_i r_i^2$2

and hence $I(t)=\sum_{i=1}^N m_i r_i^2$3 for $I(t)=\sum_{i=1}^N m_i r_i^2$4-homogeneous potentials (Cariñena et al., 2014).

A complementary formulation uses quasi-coordinates and Lie algebroids. In quasi-velocities $I(t)=\sum_{i=1}^N m_i r_i^2$5 associated with a moving frame $I(t)=\sum_{i=1}^N m_i r_i^2$6, Hamel’s coefficients $I(t)=\sum_{i=1}^N m_i r_i^2$7 encode nonholonomy through $I(t)=\sum_{i=1}^N m_i r_i^2$8. For Hamiltonian systems in quasi-coordinates, the general theorem is

$I(t)=\sum_{i=1}^N m_i r_i^2$9

for any virial function $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$0 bounded along the motion. Fiberwise-linear virial functions $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$1 are associated with complete lifts of vector fields $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$2, and the virial identity becomes $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$3. In the Lagrangian picture, if $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$4, then $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$5 and therefore

$G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$6

The same structure extends to Lie algebroids $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$7, where the virial theorem is written either as $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$8 on $G(t)=\sum_{i=1}^N \mathbf r_i\cdot \mathbf p_i.$9 or $dI/dt=2G$0 on $dI/dt=2G$1. This places the theorem in a setting adapted to systems with symmetry, moving frames, and nontrivial anchor maps (Cariñena et al., 2014).

4. Dissipative, stochastic, and open-system formulations

Dissipation modifies the dynamic virial balance by adding explicit force-virial terms from friction and noise. For the damped oscillator

$dI/dt=2G$2

the instantaneous relation is

$dI/dt=2G$3

Its long-time average gives

$dI/dt=2G$4

so the averaged kinetic and potential energies are generally unequal. For the Brownian oscillator

$dI/dt=2G$5

the dynamical identity becomes

$dI/dt=2G$6

In the stationary state, under stationarity and ergodicity, $dI/dt=2G$7 and $dI/dt=2G$8, so equipartition is restored: $dI/dt=2G$9 with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$00 when $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$01 (Ghosh et al., 2023).

For a dissipative quantum oscillator coupled to a heat bath in the Caldeira–Leggett form, the virial operator is symmetrized,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$02

and the Heisenberg equation produces bath-induced terms $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$03 and $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$04: $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$05 In the stationary state,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$06

For the harmonic oscillator this gives

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$07

so non-Markovian memory and colored quantum noise break equipartition in general. The classical limit $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$08 and the weak-coupling limit $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$09 both suppress $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$10 and $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$11, recovering the standard relation. The same structure has an electrical analogue in noisy RLC circuits, where the virial balance is written in terms of charge $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$12 and current $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$13 (Ghosh et al., 2023).

A different extension treats non-differentiable paths in resolution-scale relativity. There the complex velocity is $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$14, the scale-covariant derivative is

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$15

and the virial theorem becomes

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$16

where the quantum-like potential is

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$17

Under the identification $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$18, this reproduces the quantum mechanical virial theorem while retaining a time-average formulation rather than expectation values as the primary averaging device (LeBohec, 2022).

5. Nonequilibrium quantum gases and Tan’s contact

For short-range interacting quantum gases in three dimensions, the dynamic virial theorem acquires an exact contact term. With particle mass $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$19, scattering length $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$20, harmonic confinement $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$21, total energy $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$22, trap energy $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$23, and

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$24

the theorem reads

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$25

or equivalently

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$26

where $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$27 is Tan’s contact defined by the large-momentum tail $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$28. At unitarity, $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$29, the contact term drops out, and the cloud-size dynamics becomes identical to that of an ideal gas. The derivation uses the dilation operator $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$30, the commutator $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$31, and Tan’s adiabatic relation

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$32

The contact term is therefore the dynamical imprint of short-distance correlations in the virial balance (Peng, 2022).

This nonequilibrium theorem provides an experimentally accessible formulation of the maximum energy growth theorem. For an initially noninteracting gas with a ramp $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$33, the short-time energy growth obeys

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$34

so the square-root ramp $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$35 maximizes the initial growth. Because

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$36

during free expansion, the theorem converts measurements of cloud size and contact into a direct test of the energy-growth bound.

In two-fluid hydrodynamics, the same identity combines with continuity and momentum balance to yield an out-of-equilibrium pressure relation,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$37

where $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$38 encodes bulk dissipation. In equilibrium, with no flow and $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$39, this reduces to Tan’s pressure relation. The result is the nonequilibrium analogue of the equilibrium pressure/contact formula, expressed in terms of measurable dynamical quantities (Peng, 2022).

6. Continuum, field-theoretic, and micropolar generalizations

In continuum mechanics, the theorem is obtained by testing momentum balance against the generator of dilatations. For a material region $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$40 with density $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$41, velocity $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$42, Cauchy stress $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$43, and body-force density $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$44, define

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$45

Using Reynolds transport and Cauchy’s equation,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$46

one finds

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$47

For fluids with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$48, the stress contribution becomes $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$49. When boundary terms vanish by decay, compact support, or periodicity, the time-averaged continuum theorem reduces to the familiar bulk balance between kinetic, forcing, and trace-of-stress terms (Pommaret, 2015).

The same scaling structure appears in field theory through the dilatation Noether current

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$50

In four-dimensional Maxwell theory, the improved energy-momentum tensor is traceless,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$51

which expresses conformal invariance. The continuum virial identity is therefore the nonrelativistic projection of a dilatational balance law. One formulation further identifies the Clausius, Cosserat, Maxwell, and Weyl equations as formal adjoints of the Spencer operator for the conformal group, with the trace term arising from testing against the dilatation generator $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$52 (Pommaret, 2015).

Micropolar media introduce an additional rotational sector. With velocity $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$53, microrotation $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$54, microinertia $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$55, force stress $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$56, and couple-stress $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$57, the translational virial theorem becomes

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$58

with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$59. The rotational theorem is

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$60

with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$61. These relations uncover the virial force-stress

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$62

and the virial couple-stress

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$63

The first contains the Reynolds stress through $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$64; the second contains the turbulent couple-stress through $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$65. In the classical Cauchy limit, where microrotation and couple-stress vanish, the rotational theorem collapses and the translational theorem reduces to the standard continuum virial relation (Ostoja-Starzewski, 2021).

7. Gravitational and astrophysical variants

In Newtonian gravity, the virial theorem is central to the dynamics of self-gravitating systems. For discrete masses,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$66

and the time-dependent identity takes the form

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$67

or equivalently $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$68 in the notation used for the gravitational pair potential. For collisionless stellar systems, the continuum form is

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$69

and for self-gravity $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$70, so

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$71

In spherical collisionless systems with negative total energy, this equation governs undamped finite-amplitude breathing oscillations about virial equilibrium. One treatment maps the breathing mode to an Ermakov–Lewis–Leach structure, where $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$72 satisfies a parametric-oscillator equation and the invariant

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$73

generates an infinite asymptotic hierarchy $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$74 with $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$75 order by order. In that framework, the constants of motion depend on the virialised mass and radius through the universal scalings $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$76 and $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$77, rather than on the detailed choice of potential profile (Lecian et al., 2023).

Modified gravity changes the virial balance itself. In nonlocal Newtonian gravity, the pair force becomes

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$78

with

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$79

and the exact virial identity for an isolated $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$80-body system is

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$81

where

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$82

In virial equilibrium,

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$83

Interpreting the correction as an effective dark component leads to $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$84. For sufficiently isolated nearby galaxies in virial equilibrium at the present epoch, the theory predicts

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$85

where $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$86 is the baryonic diameter. In this setting the dynamic virial theorem is not merely diagnostic; it is the mechanism by which nonlocality is mapped onto an effective dark-matter fraction (Mashhoon, 2015).

A further model-specific complication concerns the meaning of stationarity. In a relativistic uniform sphere model with gravitational, electromagnetic, acceleration, and pressure fields, one analysis argues that the partial time derivative of the virial may vanish while the material derivative does not: $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$87 Within that framework the virial balance becomes

$$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$88

and the estimated ratio of kinetic to binding energy is $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$89, rather than the classical $$\frac{1}{2}\frac{d^2 I}{dt^2}=2K+\sum_{i=1}^N \mathbf r_i\cdot \mathbf F_i,$$90, because pressure and acceleration fields contribute to the internal dynamics. This does not alter the standard theorem as a mathematical identity; rather, it shows that the passage from stationarity to vanishing total virial derivative can depend sensitively on the kinematics and constitutive assumptions of the model under study (Fedosin, 2017).