 Rigid body dynamics

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 nondeformable, 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 significant effect on the motion of the system.
Contents
Rigid body linear momentum
Newton's Second Law states that the rate of change of the linear momentum of a particle with constant mass is equal to the sum of all external forces acting on the particle:
where m is the particle's mass, v is the particle's velocity, their product mv is the linear momentum, and f_{i} 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 has 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 with respect to time, the derivative of velocity dv/dt is the second derivative of position d^{2}r/dt^{2}, 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 r_{cm}, 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. This is known as Euler's first law.
Rigid body angular momentum
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
Given that Euler's rotation theorem states that there is always an instantaneous axis of rotation, the angular velocity, , can be given by a vector over this axis
where is a set of mutually perpendicular unit vectors fixed in a reference frame.
Rotating a rigid body is equivalent to rotating a Poinsot ellipsoid.
Angular momentum and torque
Similarly, the angular momentum for a system of particles with linear momenta p_{i} and distances r_{i} from the rotation axis is defined
For a rigid body rotating with angular velocity ω about the rotation axis (a unit vector), the velocity vector may be written as a vector cross product
where
 angular velocity vector
 is the shortest vector from the rotation axis to the point mass.
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 I is not changing) then we may write
where
 α is called the angular acceleration (or rotational acceleration) about the rotation axis .
Notice that if I is not constant in the external reference frame (i.e. the three main axes of the body are different) then we cannot take the I outside the derivate. In this cases we can have torquefree precession.
Applications
Computer physics engines use rigid body dynamics to increase interactivity and realism in video games.
See also
Theory
 Rigid body
 Rigid rotor
 Soft body dynamics
 Multibody dynamics
 Polhode
 Herpolhode
 Precession
 Poinsot's construction
Simulators
 Physics engine
 Physics processing unit
 Physics Abstraction Layer  Unified multibody simulator
 Dynamechs  Rigid body simulator
 RigidChips  Japanese rigid body simulator
External links
 Chris Hecker's Rigid Body Dynamics Information
 Physically Based Modeling: Principles and Practice
 Lectures, Computational Rigid Body Dynamics at University of WisconsinMadison
 DigitalRune Knowledge Base contains a master thesis and a collection of resources about rigid body dynamics.
Categories: Rigid bodies
 Rotational symmetry
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