Rigid body
The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it.
In quantum mechanics, a rigid body is usually thought of as a collection of point masses.
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.
[4] 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).
For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b1, b2, b3, such that b1 is parallel to the chord line of the wing and directed forward, b2 is normal to the plane of symmetry and directed rightward, and b3 is given by the cross product
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.
However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same.
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.
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:[6] 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 as the time derivative in N of its velocity:[6] For two points P and Q that are fixed on a rigid body B, where B has an angular velocity
is the position vector from P to Q.,[7] with coordinates expressed in N (or a frame with the same orientation as N.) This relation can be derived from the temporal invariance of the norm distance between P and Q.
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
[8] This equation is often combined with Acceleration of two points fixed on a rigid body.
Vehicles, walking people, etc., usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation.
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.
The configuration space of a nonfixed (with non-zero translational motion) rigid body is E+(3), the subgroup of direct isometries of the Euclidean group in three dimensions (combinations of translations and rotations).
