## 3 Natural Coordinates Formulation (NCF)
### Derivation of Generalized Forces of Rigid Body Using NCF  
In the context of the Natural Coordinates Formulation (NCF), deriving the generalized forces due to the applied forces is well established in the literature. However, the derivation of the generalized forces due to the applied moments presents greater complexity due to the absence of rotational angular coordinates in the NCF. Consequently, we begin with the derivation of the generalized forces due to moments. For the applied moments, the virtual work principle can be expressed as  
\[\delta W=\bar{\mathbf{M}}^{T}\delta^{\varsigma} \tag{10}\]  
which is the applied moment \(\bar{\mathbf{M}}\) at a given point on the body times the virtual change in orientation, \(\delta^{\varsigma}\), at that point. The virtual change \(\delta^{\varsigma}\) can be expressed as  
\[\delta^{\varsigma}=\overline{\intercal}\delta\mathbf{t} \tag{11}\]  
Therefore, the formulation of the virtual change in orientation, \(\delta^{\varsigma}\), can be obtained when the angular velocity for a material point, \(\overline{\intercal}\), along with the NCF body, is defined. Given that the coordinate set used in the NCF is similar to those employed in continuum mechanics [10], it can be concluded that the angular velocity description in the NCF aligns with the continuum mechanics framework. Thus, the vorticity tensor describes the angular velocity \(\overline{\intercal}\) at the given point in the continuum element, which can be expressed as [10]  
\[ \begin{array}{rcl}\widetilde{\mathbf{I}}&=&\begin{bmatrix}0&-\tilde{ \omega}_{z}&\tilde{\omega}_{y}\\ \tilde{\omega}_{z}&0&-\tilde{\omega}_{x}\\ -\tilde{\omega}_{y}&\tilde{\omega}_{x}&0\end{bmatrix}=\frac{1}{2}\Big{(} \mathbf{L}-\mathbf{L}^{T}\Big{)}\\ \Updownarrow\\ \Updownarrow\\ \overline{\intercal}&=&\begin{bmatrix}\tilde{\omega}_{x}\\ \tilde{\omega}_{y}\\ \tilde{\omega}_{z}\end{bmatrix}=\frac{1}{2}\begin{bmatrix}\mathbf{L}_{3,2}- \mathbf{L}_{2,3}\\ \mathbf{L}_{1,3}-\mathbf{L}_{3,1}\\ \mathbf{L}_{2,1}-\mathbf{L}_{1,2}\end{bmatrix}=\bar{\mathbf{G}}\dot{\mathbf{q}} \end{array}  \tag{13}\]  
where \(\mathbf{L}\) is the velocity gradient tensor, and \(\bar{\mathbf{G}}\) is the velocity transformation matrix. Substituting the value of \(\overline{\intercal}\) into Equation (11) and then into the virtual work principle of Equation (10) yields\[\delta W=\bar{\mathbf{M}}^{T}\bar{\mathbf{t}}\,\delta t=\bar{\mathbf{M}}^{T}\ \bar{\mathbf{G}}\ \mathbf{q}\ \delta t=\bar{\mathbf{M}}^{T}\ \bar{\mathbf{G}}\ \delta\mathbf{q} \tag{14}\]  
The velocity gradient tensor, \(\mathbf{L}\), can be defined as \(\mathbf{L}=\partial\mathbf{i}/\partial\mathbf{r}\). Thus, the generalized forces due to external moments can be described as  
\[\mathbf{Q}_{r}^{T}=\bar{\mathbf{M}}^{T}\ \bar{\mathbf{G}} \tag{15}\]  
When combining the effects of both forces and moments, the complete vector of the generalized forces takes the form  
\[\mathbf{Q}_{r}^{T}=\left(\bar{\mathbf{F}}^{T}\mathbf{C}+\bar{\mathbf{M}}^{T}\ \bar{\mathbf{G}}\right) \tag{16}\]  
In this equation, \(\bar{\mathbf{F}}=\left[\mathbf{f}_{x},\mathbf{f}_{y},\mathbf{f}_{z}\right]^{T}\) represents a concentrated force applied at action point \((x,y,z)\) on the body. The shape matrix \(\mathbf{C}\), generated using Equation (6), is evaluated at this point, capturing the force distribution and its influence on the system's configuration. The shape function \(\mathbf{C}\) ensures the correct mapping of the applied force to the generalized coordinates, maintaining consistency with the NCF framework. The equation of motion is obtained by using the method of Lagrange multipliers as  
\[ \begin{bmatrix}\mathbf{M}&\Phi_{q}^{T}\\ \Phi_{q}&0\end{bmatrix}\begin{bmatrix}\bar{\mathbf{q}}\\ \cdot\cdot\end{bmatrix} = \begin{bmatrix}\mathbf{Q}_{r}\\ \mathbf{f}\mathbf{f}\end{bmatrix}