Math 2.0

Mechanics of particles and systems Definition

Rigid body dynamics

Builds on Rigid bodyCentre of massAngular momentumTorqueAngular velocityNewton's laws of motionInertial frame of reference

Definition

Rigid body dynamics is the part of mechanics that determines the motion of rigid bodies under given forces and torques, in an inertial frame. For a rigid body made of particles of masses mi>0m_i > 0 at positions xi(t)∈ℝ3x_i(t) \in \R^3, let:

  1. m=∑imim = \sum_i m_i be the total mass and c(t)=1m∑imixi(t)c(t) = \frac{1}{m}\sum_i m_i x_i(t) the centre of mass;
  2. ω(t)\omega(t) be the angular velocity, so that x˙i=c˙+ω×(xi−c)\dot x_i = \dot c + \omega \times (x_i - c);
  3. Ic(t)=∑imi(∣ri∣2 I3−ririT)I_c(t) = \sum_i m_i \big( \abs{r_i}^2 \, \mathrm{I}_3 - r_i r_i^T \big), with ri=xi−cr_i = x_i - c and I3\mathrm{I}_3 the identity matrix, be the inertia tensor about cc;
  4. Lc=Ic ωL_c = I_c\, \omega be the angular momentum about cc.

Let external forces FkF_k act at points pkp_k. The equations of motion are the following Newton–Euler equations. m c¨=∑kFk,dLcdt=∑k(pk−c)×Fkm\, \ddot c = \sum_k F_k, \qquad \frac{d L_c}{dt} = \sum_k (p_k - c) \times F_k The second sum is the total torque about cc.

Remarks

Both equations follow from Newton's laws for the particles. One assumes that the internal forces between two particles are equal, opposite and along the line joining them, so their forces and torques cancel in pairs. Clause 4 agrees with the general definition Lc=∑imi ri×r˙iL_c = \sum_i m_i\, r_i \times \dot r_i, by the identity r×(ω×r)=∣r∣2ω−(r⋅ω)rr \times (\omega \times r) = \abs{r}^2 \omega - (r \cdot \omega) r.

Example: a body falling freely under gravity gg has total force mgm g and zero torque about cc, since gravity acts at the centre of mass. So cc follows a parabola and LcL_c is constant.

Non-example: a system whose particles move apart, such as an exploding shell, is not a rigid body. The first equation still holds for its centre of mass, but Lc=IcωL_c = I_c \omega makes no sense because there is no common angular velocity. In body-fixed principal axes the second equation becomes Euler's equations of rigid body motion.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cross product concept
  2. Derivative concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Identity matrix object
  7. Matrix definition
  8. Ordinary differential equation concept
  9. Rigid body concept
  10. Torque concept
  11. Vector space definition
  12. Velocity concept
  13. Function of several variables concept
  14. Classical mechanics concept
  15. Square matrix definition
  16. Transpose definition
  17. Linear map definition
  18. Double integral concept
  19. Momentum concept
  20. Inertial frame of reference concept
  21. Orthogonal matrix definition
  22. Multilinear map definition
  23. Angular momentum concept
  24. Triple integral concept
  25. Newton's laws of motion statement
  26. Orthogonal group object
  27. Alternating multilinear map definition
  28. Centre of mass concept
  29. Determinant definition
  30. Special orthogonal group object
  31. Angular velocity concept
  32. Rigid body dynamics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rigid_body_dynamics
About this page
TagSS3YJMRTXH
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Kinddefinition
WritingW0 stub
MSC70E15, 70E17
DefinesRigid body dynamics
UsesRigid body, Centre of mass, Angular momentum, Torque, Angular velocity, Newton's laws of motion, Inertial frame of reference
Changed3 November 2026

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