4.4. Equations of motion of a discrete system of point masses

Starting from the dynamic equations for a single point, the dynamic equations for a system of particles can be derived using the third principle of dynamics, action/reaction. The development of these equations helps us understand that the additive nature of dynamical quantities (momentum, angular momentum, kinetic energy) directly follows from their definition.

Momentum Balance. The momentum balance for each point \(i\) in the system can be written by expressing the resultant of the external forces acting on the point as the sum of the external forces acting on the entire system and the internal forces exchanged with the other points of the system,

\[\vec{R}_i^{ext,i} = \vec{F}_i^{ext} + \sum_{j \ne i} \vec{F}_{ij} \ .\]

The momentum balance equation for the \(i\)-th mass thus becomes

\[\dot{\vec{Q}}_i = \vec{R}_i^{ext,i} = \vec{F}_i^{ext} + \sum_{j \ne i} \vec{F}_{ij} \ .\]

By summing the momentum balance equations for all masses, we obtain

\[\begin{split}\begin{aligned} \sum_{i} \dot{\vec{Q}}_i & = \sum_i \vec{F}_{i}^{ext} + \sum_i \sum_{j \ne i} \vec{F}_{ij} = \\ & = \sum_i \vec{F}_{i}^{ext} + \sum_{\{i,j\}} \underbrace{\left( \vec{F}_{ij} + \vec{F}_{ji} \right)}_{=\vec{0}} \end{aligned}\end{split}\]

Defining the momentum of the system as the sum of the momenta of its parts, and the resultant of the external forces as the sum of the external forces acting on the parts of the system,

\[\vec{Q} := \sum_i \vec{Q}_i\]

\[\vec{R}^e := \sum_i \vec{F}_i^{ext}\]

we recover the general form of the momentum balance equation,

\[\dot{\vec{Q}} = \vec{R}^e \ .\]

Angular Momentum Balance. The angular momentum balance for each point \(i\) in the system can be written by expressing the resultant of the external moments acting on the point as the sum of the external moments acting on the entire system and the internal moments exchanged with the other points of the system,

\[\vec{M}_{H,i}^{ext,i} = \vec{M}_{H,i}^{ext} + \sum_{j \ne i} \vec{M}_{H,ij} \ .\]

In the case where parts of the system interact via forces, the moment with respect to a point \(H\) generated by mass \(j\) on mass \(i\) is given by

\[\vec{M}_{H,ij} = (\vec{r}_i - \vec{r}_H) \times \vec{F}_{ij} \ .\]

The angular momentum balance equation for the \(i\)-th mass thus becomes

\[\dot{\vec{L}}_{H,i} + \dot{\vec{r}}_H \times \vec{Q}_i = \vec{M}_{H,i}^{ext,i} = \vec{M}_{H,i}^{ext} + \sum_{j \ne i} \vec{M}_{H,ij} \ .\]

By summing the angular momentum balance equations for all masses, we obtain

\[\begin{split}\begin{aligned} \sum_{i} \left( \dot{\vec{L}}_i + \dot{\vec{r}}_H \times \vec{Q}_i \right) & = \sum_i \vec{M}_{H,i}^{ext} + \sum_i \sum_{j \ne i} \vec{M}_{H,ij} = \\ & = \sum_i \vec{M}_{H,i}^{ext} + \sum_{\{i,j\}} \underbrace{\left( \vec{M}_{H,ij} + \vec{M}_{H,ji} \right)}_{=\vec{0}} \end{aligned}\end{split}\]

Recognizing the total momentum of the system, and defining the angular momentum of the system as the sum of the angular momentum of its parts, and the resultant of the external moments as the sum of the external moments acting on the parts of the system,

\[\vec{L}_H := \sum_i \vec{L}_{H,i}\]

\[\vec{M}_H^e := \sum_i \vec{M}_{H,i}^{ext}\]

we recover the general form of the angular momentum balance equation,

\[\dot{\vec{L}}_{H} + \dot{\vec{r}}_H \times \vec{Q} = \vec{M}_H^e \ .\]

Kinetic Energy Balance. The kinetic energy balance of the system can be derived by taking the scalar product of the momentum balance equation for each point,

\[\vec{v}_i \cdot m_i \dot{\vec{v}}_i = \vec{v}_i \cdot \left( \vec{F}_i^{e} + \sum_{j \ne i} \vec{F}_{ij} \right) \ ,\]

recognizing in the first term the time derivative of the kinetic energy of the \(i\)-th point,

\[\dot{K}_i = \dfrac{d}{dt} \left( \frac{1}{2} m_i \vec{v}_i \cdot \vec{v}_i \right) = m_i \vec{v}_i \cdot \dot{\vec{v}}_i \ ,\]

and summing these equations to obtain

\[\begin{aligned} \sum_i \dot{K}_i = \sum_i \vec{v}_i \cdot \vec{F}_i^{e} + \sum_i \vec{v}_i \cdot \sum_{j \ne i} \vec{F}_{ij} \ . \end{aligned}\]

Defining the kinetic energy of the system as the sum of the kinetic energies of its parts, and defining the power of the external/internal forces acting on the system as the sum of the power of all external/internal forces in the system,

\[K := \sum_i K_i\]

\[P^e := \sum_i P^{ext}_i = \sum_i \vec{v}_i \cdot \vec{F}_i^{e} \]

\[P^i := \sum_i P^{int}_i = \sum_i \vec{v}_i \cdot \sum_{j \ne i} \vec{F}_{ij}\]

we recover the general form of the kinetic energy balance equation,

\[\dot{K} = P^e + P^i = P^{tot} \ .\]

Note

While internal forces and moments have zero net resultant in momentum and angular momentum balance, this is not true for power of internal actions in kinetic energy equation.