The expression of dynamical quantities for rigid bodies can be written in terms of the velocity \(\vec{v}_Q\) of a point \(Q\) of the rigid body and its angular velocity \(\vec{\omega}\), exploiting the law of rigid motion (1.1) to write the velocity of each points of the rigid system as functions of \(\vec{v}_Q\) and \(\vec{\omega}\),
3.4.1. Discrete systems
Momentum.
\[\begin{split}\begin{aligned}
\vec{Q} = \sum_k m_k \vec{v}_k
& = \sum_k m_k \left( \vec{v}_Q + \vec{\omega} \times (P_k - Q) \right) = \\
& = m \vec{v}_Q - \sum_k m_k (P_k - Q) \times \vec{\omega} = \\
& = m \vec{v}_Q + \mathbb{S}_Q \cdot \vec{\omega} \ ,
\end{aligned}\end{split}\]
having defined the static moment of inertia (a \(2^{nd}\)-order antisymmetric tensor)
\[\mathbb{S}_Q := \vec{s}_{P \times} := - \sum_k m_k (P_k - Q)_{\times} \ .\]
Angular momentum.
\[\begin{aligned}
\vec{L}_H = \sum_k (P_k-H) \times m_k \vec{v}_k
& = \underbrace{\sum_k (P_k-Q) \times m_k \vec{v}_k}_{\vec{L}_Q} + (Q - H) \times \vec{Q}
\end{aligned}\]
and
\[\begin{split}\begin{aligned}
\vec{L}_Q = \sum_k (P_k - Q) \times m_k \vec{v}_k
& = \sum_k (P_k - Q) \times m_k \left( \vec{v}_Q - (P_k - Q) \times \vec{\omega} \right) = \\
& = \mathbb{S}_Q^T \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} \ ,
\end{aligned}\end{split}\]
having recognized the transpose of the static moment of inertia, and introduced the tensor of inertia w.r.t. reference point \(Q\)
\[\mathbb{I}_Q := - \sum_k m_k \left( P_k - Q \right)_{\times} \left( P_k - Q \right)_{\times} \ .\]
Kinetic energy.
\[\begin{split}\begin{aligned}
K = \sum_k \dfrac{1}{2} m_k |\vec{v}_k|^2
& = \sum_k \dfrac{1}{2} m_k \left( \vec{v}_Q + \vec{\omega} \times (P_k - Q) \right) \cdot \left( \vec{v}_Q + \vec{\omega} \times (P_k - Q) \right) = \\
& = \sum_k \dfrac{1}{2} m_k | \vec{v}_Q |^2 + \dfrac{1}{2} \sum_k 2 m_k \vec{v}_Q \cdot \left( - (P_k - Q) \times \vec{\omega} \right) + \dfrac{1}{2} \sum_k \vec{\omega} \cdot (P_k - Q)_{\times} (P_k - Q)_\times \cdot \vec{\omega} = \\
& = \dfrac{1}{2} \left[\sum_k m_k \right] | \vec{v}_Q |^2
+ \dfrac{1}{2} \vec{v}_Q \cdot \left[ - \sum_k m_k (P_k - Q)_\times \right] \cdot \vec{\omega} + \\
& + \dfrac{1}{2} \vec{\omega} \cdot \left[ \sum_k m_k (P_k - Q)_\times \right] \cdot \vec{v}_Q
+ \dfrac{1}{2} \vec{\omega} \cdot \left[ \sum_k m_k (P_k - Q)_{\times} (P_k - Q)_\times \right] \cdot \vec{\omega} = \\
& = \dfrac{1}{2} m |\vec{v}_Q|^2 + \dfrac{1}{2} \vec{v}_Q \cdot \mathbb{S}_Q \cdot \vec{\omega} + \dfrac{1}{2} \vec{\omega} \cdot \mathbb{S}^T \cdot \vec{v}_Q + \dfrac{1}{2} \vec{\omega} \cdot \mathbb{I}_Q \cdot \vec{\omega} = \\
& = \dfrac{1}{2} \vec{v}_Q \cdot \left[ m \vec{v}_Q + \mathbb{S}_Q \cdot \vec{\omega} \right] + \dfrac{1}{2} \vec{\omega} \cdot \left[ \mathbb{S}_Q^T \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} \right] = \\
& = \dfrac{1}{2} \vec{v}_Q \cdot \vec{Q} + \dfrac{1}{2} \vec{\omega} \cdot \vec{L}_Q \ .
\end{aligned}\end{split}\]
having used the vector identities
\[\vec{a} \cdot \vec{b} \times \vec{c} = \vec{b} \cdot \vec{c} \times \vec{a}\]
\[\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}\]
and having introduced the expression of momentum and angular momentum in the last step.
Note
The components of static moment of inertia and tensor of inertia in a material basis - following the motion of the system - are constant.
3.4.2. Continuous systems
Momentum.
\[\begin{split}\begin{aligned}
\vec{Q} = \int_{V_t} \rho \vec{v}
& = \int_{V_t} \rho \left( \vec{v}_Q + \vec{\omega} \times (P - Q) \right) = \\
& = m \vec{v}_Q - \int_{V_t} \rho (P_k - Q) \times \vec{\omega} = \\
& = m \vec{v}_Q + \mathbb{S}_Q \cdot \vec{\omega} \ ,
\end{aligned}\end{split}\]
having defined the static moment of inertia (a \(2^{nd}\)-order antisymmetric tensor) for continuous systems,
\[\mathbb{S}_Q := \vec{s}_{P \times} := - \int_{V_t} \rho (P - Q)_{\times} \ .\]
Note
Using the definition of the center of mass \(G\),
\[G = \dfrac{1}{m} \int_{V_t} \rho P \ ,\]
the static moment of inertia can be written as
\[\mathbb{S}_Q = - \int_{V_t} \rho (P-Q)_{\times} = - m (G - Q)_{\times} \ .\]
Note
The components of static moment of inertia w.r.t. a material reference frame are constant. Using a material Carteisan reference frame the tensor reads
\[\begin{split}\begin{aligned}
\mathbb{S}_{Q}
& = - \int_{V_t} \rho (P - Q)_{\times} = \\
& = - \int_{V_t} \rho \left[ ( x^0 - x_Q^0 ) \hat{x}^0 + ( y^0 - y_Q^0 ) \hat{y}^0 + ( z^0 - z_Q^0 ) \hat{z}^0 \right]_\times
= S_{ij} \hat{e}^0_i \hat{e}^0_j \ ,
\end{aligned}\end{split}\]
whose components can be collected in the antisymmetric matrix
\[\begin{split}\underline{\underline{S}}_Q = \left[ S_{Q,ij} \right] = - \int_{V_0} \rho \begin{bmatrix} 0 & -(z^0-z^0_Q) & (y^0-y^0_Q) \\ (z^0 - z_Q^0) & 0 & -(x^0-x_Q^0) \\ -(y^0-y^0_Q) & (x^0-x^0_Q) & 0 \end{bmatrix} \ ,\end{split}\]
so that the vector product between \(-\int_V \rho (P-Q)\) and a vector \(\vec{a}\) reads
\[\begin{split}\begin{aligned}
- \int_{V} \rho (P-Q) \times a
& = - \int_V \rho \left[ \hat{x}^0 \left( \Delta y^0 a_z - \Delta z^0 a_y \right) + \hat{y}^0 \left( \Delta z^0 a_x - \Delta x^0 a_z \right) + \hat{z}^0 \left( \Delta x^0 a_y - \Delta y^0 a_x \right) \right] \\
& = \mathbb{S} \cdot \vec{a} \ .
\end{aligned}\end{split}\]
Angular momentum.
\[\begin{aligned}
\vec{L}_H = \int_{V_t} (P-H) \times \rho \vec{v}
& = \underbrace{\int_{V_t}(P-Q) \times \rho \vec{v}_k}_{\vec{L}_Q} + (Q - H) \times \vec{Q}
\end{aligned}\]
and
\[\begin{split}\begin{aligned}
\vec{L}_Q = \int_{V_t} (P - Q) \times \rho \vec{v}
& = \int_{V_t} (P - Q) \times \rho \left( \vec{v}_Q - (P - Q) \times \vec{\omega} \right) = \\
& = \mathbb{S}_Q^T \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} \ ,
\end{aligned}\end{split}\]
having recognized the transpose of the static moment of inertia, and introduced the tensor of inertia w.r.t. reference point \(Q\)
\[\begin{split}\begin{aligned}
\mathbb{I}_Q
& := - \int_{V_t} \rho \left( P - Q \right)_{\times} \left( P - Q \right)_{\times} = \\
& := \int_{V_t} \rho \left[ |P-Q|^2 \mathbb{I} - (P-Q) \otimes (P-Q) \right] \ ,
\end{aligned}\end{split}\]
having used the tensor identity
\[- \vec{a}_{\times} \cdot \vec{a}_{\times} = |\vec{a}|^2 \mathbb{I} - \vec{a} \otimes \vec{a}\]
Note
The components of tensor of inertia w.r.t. a material reference frame are constant. Using a material Carteisan reference frame the tensor reads
\[\begin{split}\begin{aligned}
\mathbb{I}_{Q}
& = - \int_{V_t} \rho (P - Q)_{\times} (P-Q)_{\times} = \\
& = - \int_{V_t} \rho \left[ |P-Q|^2 \mathbb{I} - (P-Q) \otimes (P-Q) \right] = \\
& = I^0_{Q,ij} \hat{e}^0_i \hat{e}^0_j \ ,
\end{aligned}\end{split}\]
whose components can be collected in the symmetric matrix
\[\begin{split}\underline{\underline{I}}^0_Q = \left[ I^0_{Q,ij} \right] = \int_{V_0} \rho \begin{bmatrix} \Delta y_0^2 + \Delta z_0^2 & -\Delta x_0 \Delta y_0 & -\Delta x_0 \Delta z_0 \\ -\Delta y_0 \Delta x_0 & \Delta x_0^2 + \Delta y_0^2 & -\Delta y_0 \Delta z_0 \\ -\Delta z_0 \Delta x_0 & - \Delta z_0 \Delta y_0 & \Delta x_0^2 + \Delta z_0^2 \end{bmatrix} \ ,\end{split}\]
being \(\Delta x^0 := x^0_P - x_Q^0\).
3.4.3. Properties of inertia tensors of rigid bodies
3.4.3.1. Static inertia
Center of mass, \(G\). Center of mass of of a rigid body is defined as the point \(G\) for which \(\mathbb{S}_G \equiv \mathbb{0}\), whose coordinates are given by
\[G = \dfrac{1}{m} \int_{V_t} \rho \, P\]
Anti-symmetric. From the definition of the static inertia tensor
\[\mathbb{S}_Q \cdot \vec{a} = \int_{V_t} \rho (P-Q) \times \vec{a} = - \vec{a} \times \int_{V_t} \rho (P-Q) = - \vec{a} \cdot \mathbb{S}_Q = - \mathbb{S}^T \cdot \vec{a} \ .\]
Transport.
\[\begin{split}\begin{aligned}
\mathbb{S}_Q
& = - \int_{V_t} \rho (P - Q)_{\times} = \\
& = - \int_{V_t} \rho (P - R)_{\times} - \int_{V_t} \rho (R - Q)_{\times} = \\
& = \mathbb{S}_R - m (R-Q)_{\times} \ ,
\end{aligned}\end{split}\]
or w.r.t. the center of mass \(G\),
\[\mathbb{S}_Q = \mathbb{S}_G - m (G-Q)_{\times} \ .\]
3.4.3.2. Tensor of inertia
Symmetric (semi)-definite positive. Inertia tensor is symmetric
\[\vec{v} \cdot \mathbb{I}_Q \cdot \vec{w} = \vec{v} \cdot \int_{V_t} \rho \left[ |\Delta \vec{r}|^2 \mathbb{I} - \Delta \vec{r} \otimes \Delta \vec{r} \right] \cdot \vec{w} = \vec{w} \cdot \mathbb{I}_Q \cdot \vec{v} \ .\]
for all \(\forall \vec{v}, \vec{w}\), and semi-definite positive
\[\begin{split}\begin{aligned}
\vec{v} \cdot \mathbb{I}_Q \cdot \vec{v}
& = - \vec{v} \cdot \int_{V_t} \rho \Delta \vec{r}_{\times} \Delta \vec{r}_{\times} \cdot \vec{v} = \\
& = - \int_{V_t} \rho \vec{v} \cdot \Delta \vec{r}_{\times} \Delta \vec{r}_{\times} \cdot \vec{v} = \\
& = - \int_{V_t} \rho \vec{v} \cdot \left[ \Delta \vec{r} \times \left( \Delta \vec{r} \times \cdot \vec{v} \right) \right] = \\
& = - \int_{V_t} \rho \left( \Delta \vec{r} \times \vec{v} \right) \cdot \left( \vec{v} \times \Delta \vec{r} \right) = \\
& = \int_{V_t} \rho \left( \Delta \vec{r} \times \vec{v} \right) \cdot \left( \Delta \vec{r} \times \vec{v} \right) = \\
& = \int_{V_t} \rho | \Delta \vec{r} \times \vec{v} |^2 \ge 0 \\
\end{aligned}\end{split}\]
Principal axes of inertia. As the tensor of inertia is symmetric and definite positive, a set of orthogonal vectors \(\hat{E}^0_i\) so that it can be written in diagonal form,
\[\mathbb{I}_Q = I^0_{XX} \, \hat{E}^0_X \otimes \hat{E}^0_X + I^0_{YY} \, \hat{E}^0_Y \otimes \hat{E}^0_Y + I^0_{ZZ} \, \hat{E}^0_Z \otimes \hat{E}^0_Z \ ,\]
with \(I_{ii}^0 \ge 0\) (no sum).
Theorem 3.1 (Transport - Huygens’ theorem.)
\[\begin{split}\begin{aligned}
\mathbb{I}_Q
& = - \int_{V_t} \rho (P - Q)_{\times} (P-Q)_{\times} = \\
& = - \int_{V_t} \rho (P - R)_{\times} (P - R)_{\times}
- \int_{V_t} \rho (P - R)_{\times} (R - Q)_{\times} \\
& \quad - \int_{V_t} \rho (R - Q)_{\times} (P - R)_{\times}
- \int_{V_t} \rho (R - Q)_{\times} (R - Q)_{\times} = \\
& = \mathbb{I}_R + \mathbb{S}_R \cdot (R-Q)_{\times} + (R-Q)_{\times} \cdot \mathbb{S}_R - m (R - Q)_{\times} (R - Q)_{\times}
\end{aligned}\end{split}\]
or w.r.t. the center of mass \(G\),
\[ \mathbb{I}_Q = \mathbb{I}_G - m (Q-G)_\times (Q-G)_\times \ .\]
3.4.4. Time derivatives of dynamical quantities
Time derivatives of dynamical quantities are easily evaluated using a Cartesian material reference frame.
Momentum.
\[\begin{split}\begin{aligned}
\dfrac{d}{dt} \vec{Q}
& = \dfrac{d}{dt} \left( m \vec{v}_Q + \mathbb{S}_Q \cdot \vec{\omega} \right) = \\
& = m \dot{\vec{v}}_Q + \dfrac{d}{dt} \left( \vec{E}^0_i S^0_{ij} \omega^0_j \right) = \\
& = m \dot{\vec{v}}_Q + \dfrac{d\vec{E}^0_i }{dt} S^0_{ij} \omega^0_j + \vec{E}^0_i S^0_{ij} \dfrac{d} \omega^0_j{dt} = \\
& = m \dot{\vec{v}}_Q + \vec{\omega} \times \vec{E}^0_i S^0_{ij} \omega^0_j + \vec{E}^0_i S^0_{ij} \dfrac{d}{dt}\omega^0_j = \\
& = m \dot{\vec{v}}_Q + \vec{\omega} \times \left( \mathbb{S}_Q \cdot \vec{\omega} \right) + \mathbb{S}_Q \cdot \dot{\vec{\omega}} \ .
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\dfrac{d}{dt} \vec{\omega}
& = \dfrac{d}{dt} \left( \hat{E}^0_i \omega_i^0 \right) = \\
& = \vec{\omega} \times \hat{E}^0_i \omega_i^0 + \hat{E}^0_i \dfrac{d \omega^0_i}{dt} = \\
& = \underbrace{\vec{\omega} \times \vec{\omega}}_{=\vec{0}} + \hat{E}^0_i \dfrac{d \omega^0_i}{dt} \ .
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\dfrac{d}{dt} \vec{v}
& = \dfrac{d}{dt} \left( \hat{E}^0_i v_i^0 \right) = \\
& = \vec{\omega} \times \hat{E}^0_i v_i^0 + \hat{E}^0_i \dfrac{d v^0_i}{dt} = \\
& = \vec{\omega} \times \vec{v} + \dfrac{{}^0 d}{dt} \vec{v} \\
\end{aligned}\end{split}\]
Angular momentum.
\[\begin{aligned}
\dfrac{d}{dt} \vec{L}_H
& = \dfrac{d}{dt} \left( (Q-H) \times \vec{Q} + \vec{L}_Q \right) \ ,
\end{aligned}\]
and
\[\dfrac{d}{dt} \left( (Q-H) \times \vec{Q} \right) = ( \vec{v}_Q - \dot{\vec{x}}_H ) \times \vec{Q} + ( Q - H ) \times \dot{\vec{Q}}\]
and
\[\begin{split}\begin{aligned}
\dfrac{d \vec{L}_Q}{dt}
& = \dfrac{d}{dt} \left( \mathbb{S}^T_Q \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} \right) = \\
& = \dfrac{d}{dt} \left[ \hat{E}_i^0 \left( S^0_{ji} v^0_{Q,j} + I^0_{ij} \omega^0_j \right) \right] = \\
& = \vec{\omega} \times \hat{E}_i^0 \left( S^0_{ji} v^0_{Q,j} + I^0_{ij} \omega^0_j \right)
+ \hat{E}^0_i \left( S_{ji}^0 \dot{v}^0_{Q,j} + I^0_{ij} \dot{\omega}^0_j \right) = \\
& = \vec{\omega} \times ( \mathbb{S}_Q^T \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} )
+ \left( \mathbb{S}_Q^T \cdot \dfrac{{}^0 d}{dt} \vec{v}_Q + \mathbb{I} \cdot \dot{\vec{\omega}} \right) = \\
& = \vec{\omega} \times ( \mathbb{S}_Q^T \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} )
+ \left( \mathbb{S}_Q^T \cdot \left( \dot{v}_Q - \vec{\omega} \times \vec{v}_Q \right) + \mathbb{I} \cdot \dot{\vec{\omega}} \right) \\
& = \left( \mathbb{S}_Q^T \cdot \left( \dot{v}_Q - \vec{\omega} \times \vec{v}_Q \right) + \mathbb{I} \cdot \dot{\vec{\omega}} \right)
+ \vec{\omega} \times ( \mathbb{S}_Q^T \cdot \vec{v}_Q + \mathbb{I}_Q \cdot \vec{\omega} )
\end{aligned}\end{split}\]
3.4.4.1. Dynamical quantities and time derivatives with \(G\) as reference point
\[\begin{split}
\begin{cases}
\vec{Q} = m \vec{v}_G \\
\vec{L}_G = \mathbb{I}_G \cdot \vec{\omega}
\end{cases}
\qquad , \qquad
\begin{cases}
\dot{\vec{Q}} = m \dot{\vec{v}}_G \\
\dot{\vec{L}}_G = \mathbb{I}_G \cdot \dot{\vec{\omega}} + \vec{\omega} \times \mathbb{I}_G \cdot \vec{\omega}
\end{cases}\end{split}\]
