Virial theorem

The virial theorem relates time-averaged kinetic energy to the forces or potential energy of a bound system. For a stable system bound by gravity with an inverse-square force, twice the average kinetic energy equals the negative average potential energy.

Virial: A scalar formed from particle positions and momenta, used to derive the virial theorem. Its time derivative yields the theorem’s central energy relation.
Galaxy cluster: A gravitationally bound system containing many galaxies, hot gas, and dark matter. Galaxy velocities and cluster size can estimate its total gravitating mass.
Hamiltonian mechanics: A formulation of classical mechanics using generalized coordinates, momenta, and a function representing total energy. Hamiltonian methods express virial relations for systems beyond simple point masses.
Rudolf Clausius: A German physicist who helped establish thermodynamics and introduced the term entropy. Clausius formulated the theorem in 1870 while studying the kinetic theory of gases.
Newton's laws of motion: Three classical laws relating motion to forces and defining inertial motion. Applying the laws to particle positions and forces produces the virial identity.
Dark matter: Matter inferred from gravitational effects that does not emit or absorb detectable light. Virial mass estimates for galaxy clusters reveal more gravitating matter than luminous matter supplies.
Inverse-square law: A law in which a force’s magnitude decreases as the square of distance increases. Newtonian gravity’s distance dependence fixes the familiar factor relating kinetic and potential energy.
Kinetic theory of gases: A theory that explains gas properties through the motions and collisions of microscopic particles. The theorem emerged from mechanics and statistical reasoning about particle motion.
Gravitational potential energy: Energy associated with the positions of masses interacting through gravity. For Newtonian gravity, its inverse-distance form gives the theorem’s familiar energy ratio.
Stellar dynamics: The study of stellar motions and gravitational structure in star systems and galaxies. The theorem links observed stellar motions to the gravitational binding of larger systems.

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