In physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom (translation in three directions), rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of freedom (translation in three directions plus rotation in three directions). Rigid bodies are also characterized as being non-deformable, as opposed to deformable bodies. As such, rigid body dynamics is used heavily in analyses and computer simulations of physical systems and machinery where rotational motion is important, but material deformation does not have a major effect on the motion of the system.
where m is the particle's mass, v is the particle's velocity, their product mv is the linear momentum, and fi is one of the N number of forces acting on the particle.
Because the mass is constant, this is equivalent to
To generalize, assume a body of finite mass and size is composed of such particles, each with infinitesimal mass dm. Each particle a position vector r. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Since velocity v is the derivative of position r, the derivative of velocity dv/dt is the second derivative of position d2r/dt2, and the linear momentum equation of any given particle is
When the linear momentum equations for all particles are added together, the internal forces sum to zero according to Newton's third law, which states that any such force has opposite magnitudes on the two particles. By accounting for all particles, the left side becomes an integral over the entire body, and the second derivative operator can be moved out of the integral, so
Let M be the total mass, which is constant, so the left side can be multiplied and divided by M, so
The expression is the formula for the position of the center of mass. Denoting this by rcm, the equation reduces to
Thus, linear momentum equations can be extended to rigid bodies by denoting that they describe the motion of the center of mass of the body.
The most general equation for rotation of a rigid body in three dimensions about an arbitrary origin O with axes x, y, z is
where the moment of inertia tensor, , is given by
and the angular velocity, , is given by
Moving any rigid body is equivalent to moving a Poinsot's ellipsoid.
Similarly, the angular momentum for a system of particles with linear momenta and distances from the rotation axis is defined
Substituting the formula for into the definition of yields
where we have introduced the special case that the position vectors of all particles are perpendicular to the rotation axis (e.g., a flywheel): .
The torque is defined as the rate of change of the angular momentum
If I is constant (because the inertia tensor is the identity, because we work in the intrinsecal frame, or because the torque is driving the rotation around the same axis so that is not changing) then we may write
Notice that if I is not constant in the external reference frame (ie. the three main axes of the body are different) then we cannot take the I outside the derivate. In this cases we can have torque-free precession.