Rigid body

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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. 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 a rigid bodies, see classification of molecules as rigid rotors.

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[edit] Rigid body kinematics

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 non-zero translational motion) rigid body is E+(n), the subgroup of direct isometries of the Euclidean group (combinations of translations and rotations).

For any point/particle of a moving (translating and rotating) body we have

\mathbf{r}(t,\mathbf{r}_0) = \mathbf{r}_c(t) + A(t) \mathbf{r}_0
\mathbf{v}(t,\mathbf{r}_0) = \mathbf{v}_c(t) + \boldsymbol\omega(t) \times (\mathbf{r}(t,\mathbf{r}_0) - \mathbf{r}_c(t)) = \mathbf{v}_c(t) + \boldsymbol\omega(t) \times A(t) \mathbf{r}_0
\dot{A}(t)\mathbf{r}_0 = \boldsymbol\omega(t) \times A(t)\mathbf{r}_0

where

  • \mathbf{r}(t) is the position of the point/particle at time t\,
  • \mathbf{r}_c(t) is the position of a reference point of the body at time t\,
  • A(t)\, is the orientation, an orthogonal matrix with determinant 1
  • \dot{A}(t)\, is the time derivative of A(t),\, see Newton's notation for differentiation. The derivative of a matrix is a matrix with the derivative of each component (think of the matrix as three vectors, one in each column)
  • \mathbf{r}_0 is the position of the point/particle with respect to the reference point of the body in a reference orientation, for instance \mathbf{r}_0 =\mathbf{r}(0)\, (the reference orientation is the one at initial time)
  • \boldsymbol\omega is the angular velocity
  • \mathbf{v} is the total velocity of the point/particle
  • \mathbf{v}_c is the translational velocity

[edit] Dynamics of the rigid body

Main article: Rigid body dynamics

To describe the motion, the reference point \mathbf{r}_0 can be any point that is rigidly connected to the body (the translation vector depends on the choice). Depending on the application a convenient choice may be:

  • the center of mass of the whole system; properties:
    • the (linear) momentum: the total mass of the rigid body times the translational velocity. The net external force on the rigid body is the total mass times the translational acceleration (i.e., Newton's second law holds for the translational motion). The linear momentum is independent of the rotational motion.
    • the angular momentum with respect to the center of mass is the same as without translation: at any time it is equal to the inertia tensor times the angular velocity. When the angular velocity is expressed with respect to the principal axes frame of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; the torque is the inertia tensor times the angular acceleration.
    • possible motions in the absence of external forces are translation with constant velocity, steady rotation about a fixed principal axis, and also torque-free precession.
    • the total kinetic energy is simply the sum of translation and rotational energy
  • a point such that the translational motion is zero or simplified, e.g on an axle or hinge, at the center of a ball-and-socket joint, etc.

[edit] Matricial notation of rotations

When the cross product

\boldsymbol\omega(t) \times A(t)\mathbf{r}_0

is written as a matrix multiplication, this matrix is a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements,

\boldsymbol\omega(t) \times A(t)\mathbf{r}_0 = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0 & -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix} A(t)\mathbf{r}_0.

In 2D the matrix A(t) simply represents a rotation in the xy-plane by an angle which is the integral of the scalar 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.

The orientation can also be described in a different way, e.g. as a unit-quaternion-valued function of time. Although the latter is specific up to a factor -1, it would be reasonable to choose it continuously.

Two rigid bodies are said to be different (not copies) is that 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 S2n, 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:

  • the sheet surface with the image is not symmetric - in this case the two sides are different, but the mirror image of the object is the same, after a rotation by 180° about the axis perpendicular to the mirror plane.
  • the sheet surface with the image has a symmetry axis - in this case the two sides are the same, and the mirror image of the object is also the same, again after a rotation by 180° about the axis perpendicular to the mirror plane.

A sheet with a through and through image is achiral. We can distinguish again two cases:

  • the sheet surface with the image has no symmetry axis - the two sides are different
  • the sheet surface with the image has a symmetry axis - the two sides are the same

[edit] See also

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