Rigid body dynamics

In physics, rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate, which introduces other considerations.

Note: This article has much overlap with the rigid rotor and rigid body articles. Articles should eventually be merged.

Equations from particle dynamics can be generalized to rigid body dynamics as follows:

Rigid body linear momentum

The equation for particle linear momentum is

where:
• m is the particle's mass.
• v is the particle's velocity.
• fi is one of the N forces acting on the particle.
Assuming constant mass, this reduces to

To generalize assume a body of finite mass and size is composed of such particles. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Each particle has:
• a mass .
• a position vector r.
Thus, the linear momentum equation of any given particle would look like this:

If the equation for each particle were added together, the internal forces would cancel out, since by Newton's third law, any such force would have opposite magnitudes on the two particles. Also, the left side would become an integral over the entire body, and the second derivative operator could come out of the integral, leaving

Letting M be the total mass, the left side can be multiplied and divided by M without changing the validity:

However, is the formula for the position of 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.

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:
Here is the moment of inertia tensor and is the angular velocity (a vector). Based on this, a theorem states that any rigid body is equivalent when moving to a Poinsot's ellipsoid.

Further
• ωq is the angular velocity about axis q.
• M is the total mass.
• bG/O is the vector from O to the body's center of mass.
• RO is the position of O.
• t is time.
• is an integral over the mass of the body.
• is one of the N moments about O.
This equation follows from equation for linear momentum of a particle and kinematics; no additional observations of nature are necessary to arrive at it.

There are many special cases that simplify this equation. The first term goes to zero if any of three conditions are met:
• O is a fixed point (since its second derivative would be zero).
• A set of axes is chosen with its origin attached to the body's center of mass (since this would reduce the vector b to zero).
• The vector b always points in the direction of the acceleration of O (since the cross product of parallel vectors is zero).
Also, if the axes are chosen are the principal axes (i.e., the moments about the xy, xz, and yz planes is zero), the off-diagonal terms of the matrix are zero. This case is further discussed by Euler's equations.

When learning about angular motion, students are generally first exposed to the case of rotation only in the x-y plane and a fixed axis or axis at the center of mass with constant rotational inertia. That equation is

Angular momentum and torque

Similarly, the angular momentum for a system of particles with linear momenta and distances from the rotation axis is defined

For a rigid body rotating with angular velocity 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 is not changing) then we may write

where
is called the angular acceleration (or rotational acceleration) about the rotation axis .

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.

Applications

Computer physics engines use rigid body dynamics to increase interactivity and realism in video games.

Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it.
In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. The former distinguishes it from kinematics and the latter distinguishes it from statics.
The term SPACE (capitalized) can refer to:
• , a Canadian science-fiction channel
• The Society for Promotion of Alternative Computing and Employment
• DSPACE, a term in computational complexity theory

A rotation is a movement of an object in a circular motion.
momentum (pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.
center of mass of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The center of mass is a function only of the positions and masses of the particles that comprise the system.

Moment of inertia, also called mass moment of inertia and, sometimes, the
angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed at which an object is rotating along with the direction in which it is rotating.
In classical mechanics, Poinsot's construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting.
Euler's equations describe the rotation of a rigid body in a frame of reference fixed in the rotating body

where are the applied torques, are the principal moments of inertia and are the components of the angular velocity vector along the principal
angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1 (the unit length). A unit vector is often written with a superscribed caret or “hat”, like this (pronounced "i-hat").
cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result.
flywheel is a rotating disk used as a storage device for kinetic energy. Flywheels resist changes in their rotational speed, which helps steady the rotation of the shaft when a fluctuating torque is exerted on it by its power source such as a piston-based (reciprocating) engine, or
torque (or often called a moment) can informally be thought of as "rotational force" or "angular force" which causes a change in rotational motion. This force is defined by linear force multiplied by a radius.

The SI unit for torque is the newton meter (N m). In U.S.
Precession refers to a change in the direction of the axis of a rotating object. In physics, there are two types of precession, torque-free and torque-induced, the latter being discussed here in more detail.
A physics engine is a computer program that simulates Newtonian physics models, using variables such as mass, velocity, friction and wind resistance. It can simulate and predict effects under different conditions that would approximate what happens in real life or in a fantasy
video game is a game that involves interaction with a user interface to generate visual feedback on a video device.

The word video in video game traditionally refers to a raster display device.
rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it.