4.3. Equations of motion of a point mass

Dynamic quantities.

\[\begin{split}\begin{aligned} \vec{Q}_P & := m_P \vec{v}_P \\ \vec{L}_{P,H} & := (\vec{r}_P - \vec{r}_H) \times \vec{Q} = m_P (\vec{r}_P - \vec{r}_H) \times \vec{v}_P \\ K & := \frac{1}{2} m_P \vec{v}_P \cdot \vec{v}_P = \frac{1}{2} m_P |\vec{v}_P|^2 \end{aligned}\end{split}\]

Momentum balance equation. The balance equation of momentum of a point \(P\) with mass \(m\), \(\vec{Q}_P = m \vec{v}_P\) readily follows the second principle of dynamics,

\[\dot{\vec{Q}}_P = \vec{R}^e_P\]

Angular momentum balance equation. Time derivative of the angular momentum is evaluated with the rule of derivative of product,

\[\begin{split}\begin{aligned} \dot{\vec{L}}_{P,H} & = \dfrac{d}{dt} \left[ m_P (\vec{r}_P - \vec{r}_H) \times \vec{v}_P \right] = \\ & = m \left[ ( \dot{\vec{r}}_P - \dot{\vec{r}}_H ) \times \vec{v}_P + m_P (\vec{r}_P - \vec{r}_H) \times \dot{\vec{v}}_P \right] = \\ & = - m_P \dot{\vec{r}}_H \times \vec{v}_P + m_P (\vec{r}_P - \vec{r}_H) \times \dot{\vec{v}}_P = \\ & = - \dot{\vec{r}}_H \times \vec{Q} + \vec{M}_H^{ext} \ . \end{aligned}\end{split}\]

Kinetic energy blanace equation.

\[\begin{split}\begin{aligned} \dot{K}_{P} & = \dfrac{d}{dt} \left( \frac{1}{2} m_P \vec{v}_P \cdot \vec{v}_P \right) = \\ & = m_P \dot{\vec{v}}_P \cdot \vec{v}_P = \\ & = \vec{R}^e \cdot \vec{v}_P = P^e = P^{tot} \\ \end{aligned}\end{split}\]

being the power of external actions \(P^e\) equal to the total power acting on the system, assuming there is no internal action in the point system, or at least they have zero net power.