right|thumb|The position of a rigid body is determined by the position of its center of mass and by its [[Attitude (geometry)|attitude (at least six parameters in total).]]

In classical mechanics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible, when a deforming pressure or deforming force is applied on it. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass. Mechanics of rigid bodies is a field within mechanics where motions and forces of objects are studied without considering effects that can cause deformation (as opposed to mechanics of materials, where deformable objects are considered).

In the study of special relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light, where the mass is infinitely large. In quantum mechanics, a rigid body is usually thought of as a collection of point masses. For instance, molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors).

Principles

Linear and angular position

The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:

  1. the linear position or position of the body, namely the position of one of the particles of the body, specifically chosen as a reference point (typically coinciding with the center of mass or centroid of the body), together with
  2. the angular position (also known as orientation, or attitude) of the body.

Thus, the position of a rigid body has two components: linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as linear and angular velocity, acceleration, momentum, impulse, and kinetic energy.<!--

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The linear position can be represented by a vector with its tail at an arbitrary reference point in space (the origin of a chosen coordinate system) and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its center of mass or centroid. This reference point may define the origin of a coordinate system fixed to the body.

There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix (also referred to as a rotation matrix). All these methods actually define the orientation of a basis set (or coordinate system) which has a fixed orientation relative to the body (i.e. rotates together with the body), relative to another basis set (or coordinate system), from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b<sub>1</sub>, b<sub>2</sub>, b<sub>3</sub>, such that b<sub>1</sub> is parallel to the chord line of the wing and directed forward, b<sub>2</sub> is normal to the plane of symmetry and directed rightward, and b<sub>3</sub> is given by the cross product <math> b_3 = b_1 \times b_2 </math>.

In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).

Linear and angular velocity

Velocity (also called linear velocity) and angular velocity are measured with respect to a frame of reference.

The linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation.

Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating (the existence of this instantaneous axis is guaranteed by the Euler's rotation theorem). All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation. The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity.

Kinematical equations

Addition theorem for angular velocity

The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D:

:<math> {}^\mathrm{N}\!\boldsymbol{\omega}^\mathrm{B} = {}^\mathrm{N}\!\boldsymbol{\omega}^\mathrm{D} + {}^\mathrm{D}\!\boldsymbol{\omega}^\mathrm{B}.</math>

In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.

Addition theorem for position

For any set of three points P, Q, and R, the position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R:

:<math> \mathbf{r}^\mathrm{PR} = \mathbf{r}^\mathrm{PQ} + \mathbf{r}^\mathrm{QR}.</math>

The norm of a position vector is the spatial distance.

Here the coordinates of all three vectors must be expressed in coordinate frames with the same orientation.

Mathematical definition of velocity

The velocity of point P in reference frame N is defined as the time derivative in N of the position vector from O to P:

:<math> {}^\mathrm{N}\mathbf{v}^\mathrm{P} = \frac{\mathrm{d}t}(\mathbf{r}^\mathrm{OP}) </math>

where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/dt operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N.

Mathematical definition of acceleration

The acceleration of point P in reference frame N is defined as the time derivative in N of its velocity:

:<math> {}^\mathrm{N}\mathbf{v}^\mathrm{Q} = {}^\mathrm{N}\!\mathbf{v}^\mathrm{P} + {}^\mathrm{N}\boldsymbol{\omega}^\mathrm{B} \times \mathbf{r}^\mathrm{PQ}.</math>

where <math>\mathbf{r}^\mathrm{PQ} </math> is the position vector from P to Q., This relation is often combined with the relation for the Velocity of two points fixed on a rigid body.

Acceleration of one point moving on a rigid body

The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by

:<math> {}^\mathrm{N}\mathbf{a}^\mathrm{R} = {}^\mathrm{N}\mathbf{a}^\mathrm{Q} + {}^\mathrm{B}\mathbf{a}^\mathrm{R} + 2 {}^\mathrm{N}\boldsymbol{\omega}^\mathrm{B} \times {}^\mathrm{B}\mathbf{v}^\mathrm{R} </math>

where Q is the point fixed in B that instantaneously coincident with R at the instant of interest.