Mechanics of particles and systems Foundations of particle mechanics Theorem

Virial theorem

Builds on Newton's laws of motion Kinetic energy Potential energy Fundamental theorem of calculus

Statement

Let $N$ point particles have masses $m_i > 0$ and positions $x_i(t) \in \R^3$ for $t \ge 0$ , and let their motion obey Newton's second law with net forces $F_i(t)$ . Suppose that the positions and velocities stay bounded for $t \ge 0$ . For a function $f$ of $t$ , write $\langle f \rangle = \lim_{\tau \to \infty} \frac{1}{\tau}\int_0^\tau f(t)\, dt$ when the limit exists. Let $T = \sum_i \frac12 m_i \abs{\dot x_i}^2$ be the kinetic energy.

If one of $\langle T \rangle$ and $\langle \sum_i F_i \cdot x_i \rangle$ exists, then so does the other, and $2 \langle T \rangle = -\big\langle \sum_{i=1}^{N} F_i \cdot x_i \big\rangle$ .
Let $U \subseteq \R^{3N}$ be an open set that contains the motion, with $\lambda U = U$ for all $\lambda > 0$ . Suppose moreover that $F_i = -\nabla_{x_i} V$ for a potential energy $V \colon U \to \R$ of class $C^1$ . Suppose that $V(\lambda x) = \lambda^k V(x)$ for all $\lambda > 0$ and $x \in U$ . Then $2 \langle T \rangle = k \langle V \rangle$ , in the same sense as in part 1.

Remarks

Let $G(t) = \sum_i m_i \dot x_i \cdot x_i$ . Then $\dot G = 2T + \sum_i F_i \cdot x_i$ , and $\frac1\tau \int_0^\tau \dot G\, dt = (G(\tau) - G(0))/\tau \to 0$ , since $G$ is bounded; this gives part 1 by the fundamental theorem of calculus. For part 2, Euler's identity for homogeneous functions gives $\sum_i x_i \cdot \nabla_{x_i} V = k V$ , so $\sum_i F_i \cdot x_i = -kV$ . Example: for the Kepler problem, $k = -1$ and $2\langle T \rangle = -\langle V \rangle$ ; for a harmonic oscillator, $k = 2$ and $\langle T \rangle = \langle V \rangle$ . Boundedness is needed: for a free particle $x(t) = t v$ with $v \ne 0$ , one has $F = 0$ but $\langle T \rangle = \frac12 m \abs{v}^2 \ne 0$ .

Going deeper

For a student

The theorem relates averages without solving the equations of motion, which makes it useful for many-body systems. Astronomers use it to estimate the masses of star clusters and galaxies from the speeds of their members. For bound Kepler orbits it combines with energy conservation to give $\langle T \rangle = -E$ and $\langle V \rangle = 2E$ , where $E$ is the energy.

Reading path

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Derivative concept
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Ordinary differential equation concept
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Velocity concept
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Ordinary differential equation definition
Function of several variables concept
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Lipschitz condition definition
Classical mechanics concept
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Local minimum definition
Mesh of a partition definition
Riemann sum definition
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Limit of a function of several variables concept
Momentum concept
Partial derivative concept
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Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Continuity in R^n definition
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Partial derivative definition
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Newton's laws of motion statement
Continuous function definition
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Fermat's theorem on stationary points statement
Riemann integral definition
Convergent sequence definition
Monotone convergence theorem for sequences statement
Every sequence has a monotone subsequence statement
Differentiable on an open set definition
Gradient definition
Higher partial derivatives definition
Picard–Lindelöf hypotheses definition
Intermediate value theorem statement
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Iterated derivative definition
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Additivity of the integral over intervals statement
Boundedness theorem statement
Monotonicity of the integral statement
C^k function characterised by continuous derivatives statement
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Extreme value theorem statement
Vector field definition
Picard–Lindelöf theorem statement
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Rolle's theorem statement
Conservative vector field definition
Mean value theorem statement
First fundamental theorem of calculus statement
Classical mechanics definition
Potential energy definition
Momentum definition
Second fundamental theorem of calculus statement
Kinetic energy definition
Fundamental theorem of calculus statement
Virial theorem statement

Tag	`6T0BTVNMRF`
URI	`m2://6T0BTVNMRF`
Kind	statement, theorem
Writing	W0 stub
MSC	70F10, 70A05
Status	This copy of the site has no build data. Run `m2 status 6T0BTVNMRF`.
About	Kinetic energy, Potential energy
Uses	Newton's laws of motion, Kinetic energy, Potential energy, Fundamental theorem of calculus
Same as	Wikidata Q620602 Wikipedia Virial theorem
Changed	3 November 2026

Virial theorem

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