Classical mechanics  
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In physics, a 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. Even though such an object cannot physically exist due to relativity, objects can normally be assumed to be perfectly rigid if they are not moving near the speed of light.
In classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors).
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 noncollinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their timeinvariant position relative to the three selected particles is known. However, typically a different and mathematically more convenient approach is used. The position of the whole body is represented by:
Thus, the position of a rigid body has two components: linear and angular, respectively.^{[2]} The same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as velocity, acceleration, momentum, impulse, and kinetic energy. ^{[3]}
The linear position can be represented by a vector with its tail at an arbitrary reference point in space (often the origin of a chosen coordinate system) and its tip at a point of interest on the rigid body (often its center of mass or centroid).
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).
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 (rototranslation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).
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 lay 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. 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.
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 D^{[4]}:
In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.
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:
The velocity of point P in reference frame N is defined using the time derivative in N of the position vector from O to P^{[5]}:
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.
The acceleration of point P in reference frame N is defined using the time derivative in N of its velocity^{[5]}:
.
For two points P and Q that are fixed on a rigid body B, where B has an angular velocity in the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N^{[6]}:
By differentiating the equation for the Velocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as
where is the angular acceleration of B in the reference frame N^{[6]}.
If the point R is moving in rigid body B while B moves in reference frame N, then the velocity of R in N is
where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest^{[7]}. This relation is often combined with the relation for the Velocity of two points fixed 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
where Q is the point fixed in B that instantaneously coincident with R at the instant of interest^{[7]}. This equation is often combined with Acceleration of two points fixed on a rigid body.
If C is the origin of a local coordinate system L, attached to the body,
where
In 2D the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the xyplane by an angle which is the integral of the angular velocity over time.
Vehicles, walking people, etc. usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is the winding number with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon.
Any point that is rigidly connected to the body can be used as reference point (origin of coordinate system L) to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on the choice).
However, depending on the application, a convenient choice may be:
When the center of mass is used as reference point:
Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the other. A rigid body is called chiral if its mirror image is different in that sense, i.e., if it has either no symmetry or its symmetry group contains only proper rotations. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version. The latter applies for S_{2n}, of which the case n = 1 is inversion symmetry.
For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases:
A sheet with a through and through image is achiral. We can distinguish again two cases:
The configuration space of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying manifold of the rotation group SO(3). The configuration space of a nonfixed (with nonzero translational motion) rigid body is E^{+}(3), the subgroup of direct isometries of the Euclidean group in three dimensions (combinations of translations and rotations).
