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: : {}^\mathrm{N}\!\boldsymbol{\omega}^\mathrm{B} = {}^\mathrm{N}\!\boldsymbol{\omega}^\mathrm{D} + {}^\mathrm{D}\!\boldsymbol{\omega}^\mathrm{B}. 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: : \mathbf{r}^\mathrm{PR} = \mathbf{r}^\mathrm{PQ} + \mathbf{r}^\mathrm{QR}. 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: : {}^\mathrm{N}\mathbf{v}^\mathrm{P} = \frac{{}^\mathrm{N}\mathrm{d}}{\mathrm{d}t}(\mathbf{r}^\mathrm{OP}) where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/d
t 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: : {}^\mathrm{N}\mathbf{v}^\mathrm{Q} = {}^\mathrm{N}\!\mathbf{v}^\mathrm{P} + {}^\mathrm{N}\boldsymbol{\omega}^\mathrm{B} \times \mathbf{r}^\mathrm{PQ}. where \mathbf{r}^\mathrm{PQ} 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 : {}^\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} where Q is the point fixed in B that instantaneously coincident with R at the instant of interest. This equation is often combined with
Acceleration of two points fixed on a rigid body.
Other quantities If
C is the origin of a local
coordinate system L, attached to the body, the
spatial or
twist acceleration of a rigid body is defined as the
spatial acceleration of
C (as opposed to material acceleration above): \boldsymbol\psi(t,\mathbf{r}_0) = \mathbf{a}(t,\mathbf{r}_0) - \boldsymbol\omega(t) \times \mathbf{v}(t,\mathbf{r}_0) = \boldsymbol\psi_c(t) + \boldsymbol\alpha(t) \times A(t) \mathbf{r}_0 where • \mathbf{r}_0 represents the position of the point/particle with respect to the reference point of the body in terms of the local coordinate system
L (the rigidity of the body means that this does not depend on time) • A(t)\, is the
orientation matrix, an
orthogonal matrix with determinant 1, representing the
orientation (angular position) of the local coordinate system
L, with respect to the arbitrary reference orientation of another coordinate system
G. Think of this matrix as three orthogonal unit vectors, one in each column, which define the orientation of the axes of
L with respect to
G. • \boldsymbol\omega(t) represents the
angular velocity of the rigid body • \mathbf{v}(t,\mathbf{r}_0) represents the total velocity of the point/particle • \mathbf{a}(t,\mathbf{r}_0) represents the total acceleration of the point/particle • \boldsymbol\alpha(t) represents the
angular acceleration of the rigid body • \boldsymbol\psi(t,\mathbf{r}_0) represents the
spatial acceleration of the point/particle • \boldsymbol\psi_c(t) represents the
spatial acceleration of the rigid body (i.e. the spatial acceleration of the origin of
L). In 2D, the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the
xy-plane 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.
Instantaneous rotation axis formulae Assume that \mathbf v(\mathbf P) is a smooth 3-d vector field and O is a point in \mathbb R^3, with \mathbf v_O = \mathbf v(O) . Denote B_\varepsilon the ball of radius \varepsilon centered at O, and \mathbf r = \mathbf P - O . We examine the expression \mathbf I_\varepsilon = \int_{B_\epsilon} \frac{\mathbf r \times (\mathbf v(\mathbf P) - \mathbf v_O)}{r^2} \, dV. Linearizing the velocity field at O gives \mathbf v(\mathbf P) - \mathbf v_O = (\nabla \mathbf v)_O \, \mathbf r + o(r), where (\nabla \mathbf v)_O is the Jacobian matrix at O. Decompose it into symmetric and antisymmetric parts: (\nabla \mathbf v)_O = J_s + J_a, with J_a antisymmetric. By linear algebra, there exists a vector \boldsymbol{w} such that J_a \mathbf r = \boldsymbol{w} \times \mathbf r. In fact, direct computation shows that \boldsymbol w = {1\over 2} \nabla \times \mathbf v(O). The symmetric part J_s does not contribute to the integral, hence \mathbf I_\varepsilon = \int_{B_\varepsilon} \frac{\mathbf r \times (J_a \mathbf r)}{r^2} \, dV + o(\varepsilon^3) = \int_{B_\varepsilon} \frac{\mathbf r \times (\boldsymbol{w} \times \mathbf r)}{r^2} \, dV + o(\varepsilon^3). Using the triple product identity, there holds \frac{\mathbf r \times (\boldsymbol{w} \times \mathbf r)}{r^2} = \boldsymbol{w} - \frac{(\mathbf r \cdot \boldsymbol{w}) \mathbf r}{r^2}. Integrating over the ball and using spherical symmetry, \int_{B_\varepsilon} \frac{(\mathbf r \cdot \boldsymbol{w}) \mathbf r}{r^2} \, dV = \frac{1}{3} \text{Vol}(B_\varepsilon) \boldsymbol{w}, so that \mathbf I_\varepsilon = \frac{2}{3} \text{Vol}(B_\varepsilon) \boldsymbol{w} + o(\varepsilon^3), \quad {\rm with} \quad \boldsymbol w = {1\over 2} \nabla \times \mathbf v(O).\quad (*) Incidentally, this formula provides an integral formulation of the curl of the vector field at O: \nabla \times \mathbf v(O) = \lim_{\varepsilon \to 0} {3\over \text{Vol}(B_\varepsilon)} \int_{B_\epsilon} \frac{\mathbf r \times (\mathbf v(\mathbf P) - \mathbf v_O)}{r^2} \, dV.
Coordinate free formula for the instantaneous rotation vector Now, assume a rigid body is rotating with angular velocity \boldsymbol{\omega} . By rigid body kinematics, using the notations above, the field of velocities is given at every time t by \mathbf v(\mathbf P) = \mathbf v_O + \boldsymbol{\omega} \times \mathbf r. Thus, the vector field \mathbf v(\mathbf \mathbf P) - \mathbf v_O is linear in \mathbf r. It follows that (\nabla \mathbf v)_O \, \mathbf r = \boldsymbol\omega \times \mathbf r = J_a. Thus \boldsymbol\omega = \boldsymbol w and the terms o(r) and o(\varepsilon^3) vanish identically in the above formulae. Therefore (*) implies \mathbf I_\varepsilon = \frac{2}{3} \mathrm{Vol}(B_\varepsilon)\, \boldsymbol{\omega}. Solving for \boldsymbol{\omega} yields, for every ball B_\varepsilon centered at O, \boldsymbol{\omega} = \frac{3}{2\,\mathrm{Vol}(B_\varepsilon)} \int_{B_\varepsilon} \frac{\mathbf r \times (\mathbf v(\mathbf P) - \mathbf v_O)}{r^2} \, dV.
Curl formula From (*) and the fact that o(\varepsilon^3) vanishes identically (as seen just above), the curl formula follows: \boldsymbol{\omega} = \frac{1}{2} \nabla \times \mathbf v(O). ==Kinetics==