Three-body inertia tensor

June Haak Ee, Dong Won Jung, U. Rae Kim, Dohyun Kim, Jungil Lee

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


We derive a general formula for the inertia tensor of a rigid body consisting of three particles with which students can learn basic properties of the inertia tensor without calculus. The inertia-tensor operator is constructed by employing the Dirac's bra-ket notation to obtain the inertia tensor in an arbitrary frame of reference covariantly. The principal axes and moments of inertia are computed when the axis of rotation passes the center of mass. The formulas are expressed in terms of the relative displacements of particles that are determined by introducing Lagrange's undetermined multipliers. This is a heuristic example analogous to the addition of a gauge-fixing term to the Lagrangian density in gauge field theories.We confirm that the principalmoments satisfy the perpendicularaxis theorem of planar lamina. Two special cases are considered as pedagogical examples. One is a water-molecule-like system in which a particle is placed on the vertical bisector of two identical particles. The other is the case in which the center of mass coincides with the incenter of the triangle whose vertices are placed at the particles. The principal moment of the latter example about the normal axis is remarkably simple and proportional to the product 'abc' of the three relative distances. We expect that this new formula can be used in actual laboratory classes for general physics or undergraduate classical mechanics.

Original languageEnglish
Article number055016
JournalEuropean Journal of Physics
Issue number5
Publication statusPublished - 2021 Sept


  • Gauge fixing
  • Inertia tensor
  • Lagrange's multiplier
  • Principal axes
  • Three-body system

ASJC Scopus subject areas

  • Physics and Astronomy(all)


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