Mathematics is a language”
From the d’Alembert’s principle, Gauss provided an ingenious reinterpretation of the d’Alembert’s principle [1]. Recall the d’Alembert’s principle: \[ \sum_{i=1}^{N} ( \mathbf{F}_i - m_i\ddot{\mathbf{r}}_i ) \cdot \delta \mathbf{r}_i = 0 \tag{1} \] The position of the \(i\)-th particle \(\mathbf{r}_i\) is a function of \(n\) generalized coordinates, \(q_1, q_2, \cdots, q_n\): \[ \mathbf{r}_i = \mathbf{r}_i(q_1, q_2, \cdots, q_n,t ), ~~~~~ i = 1, 2, \cdots, N \] Recall that we have derived the cancellation of dots identity from the time derivative of \(\mathbf{r}_i\): \[ \dot{\mathbf{r}}_i = \sum_{j=1}^{n}\frac{\partial \mathbf{r}_i}{\partial q_j }\dot{q}_j+ \frac{\partial \mathbf{r}_i}{\partial t}, ~~~~~~~~~~~~~~~~ \frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j} \] We can again take the time derivative of \(\dot{\mathbf{r}}_i\) and derive [2]:1 \[ \ddot{\mathbf{r}}_i = \sum_{j=1}^{n} \sum_{k=1}^{n} \frac{\partial^2 \mathbf{r}_i}{\partial q_j \partial q_k}\dot{q}_j\dot{q}_k + \sum_{j=1}^{n}\frac{\partial \mathbf{r}_i}{\partial q_j }\ddot{q}_j + \sum_{j=1}^{n}\frac{\partial^2 \mathbf{r}_i}{\partial q_j \partial t} + \frac{\partial^2 \mathbf{r}_i}{\partial t^2} ~~~~~~~~~~~~~ \frac{\partial \ddot{\mathbf{r}}_i}{\partial \ddot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j} \tag{2} \] Using Equation 2 in one of the terms of d’Alembert’s principle (Equation 1): \[ \begin{align*} \sum_{i=1}^{N} m_i\ddot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i &= \sum_{i=1}^{N} m_i\ddot{\mathbf{r}}_i \cdot \Big( \sum_{j=1}^{n}\frac{\partial \mathbf{r}_i}{\partial q_j}\delta q_j \Big) = \sum_{i=1}^{N} m_i\ddot{\mathbf{r}}_i \cdot \Big( \sum_{j=1}^{n}\frac{\partial \ddot{\mathbf{r}}_i}{\partial \ddot{q}_j}\delta q_j \Big) \\ &= \sum_{j=1}^{n} \Big( \sum_{i=1}^{N} m_i \ddot{\mathbf{r}}_i \frac{\partial \ddot{\mathbf{r}}_i}{\partial \ddot{q}_j} \Big) \delta q_j = \sum_{j=1}^{n}\Big( \frac{\partial Z}{\partial \ddot{q}_j} \Big) \delta q_j \end{align*} \] In these equations, \(Z\) is the energy of acceleration [2], defined by: \[ Z := \sum_{i=1}^{N}\frac{1}{2}m_i \ddot{\mathbf{r}}_i \cdot \ddot{\mathbf{r}}_i = \sum_{i=1}^{N}\frac{1}{2}m_i ||\ddot{\mathbf{r}}_i ||^2 \] By using generalized forces \(Q_1, Q_2, \cdots, Q_n\) conjugate to the generalized coordinates \(q_1, q_2, \cdots, q_n\) and \(\delta q_1, \delta q_2, \cdots \delta q_n\) are independent, we have: \[ \sum_{i=1}^{N} ( \mathbf{F}_i - m_i\ddot{\mathbf{r}}_i ) \cdot \delta \mathbf{r}_i = \sum_{j=1}^{n}\Big( Q_j -\frac{\partial Z}{\partial \ddot{q}_j} \Big) \delta q_j = 0 ~~~~~~~~~~~~~~ Q_j - \frac{\partial Z}{\partial \ddot{q}_j} = 0, ~~~~ j = 1, 2, \cdots, n \] The result can be considered as given \(q_1, q_2, \cdots, q_n\) and \(\dot{q}_1, \dot{q}_2, \cdots \dot{q}_n\), we minimize the following function as a function of \(\ddot{q}_1, \ddot{q}_2, \cdots \ddot{q}_n\): \[ f(\ddot{q}_1, \ddot{q}_2, \cdots, \ddot{q}_n) = \sum_{j=1}^{n} Q_j \ddot{q}_j- Z, ~~~~~~~~~ \frac{\partial f}{\partial \ddot{q}_j} = 0, ~~j = 1, 2, \cdots, n \] This is the Appell’s equation of motion
Note that \(\dot{\mathbf{r}}_i=\dot{\mathbf{r}}_i(q_1, q_2, \cdots q_n, \dot{q}_1, \dot{q}_2, \cdots, \dot{q}_n , t)\) and \(\frac{\partial \mathbf{r}_i}{\partial q_j}=\frac{\partial \mathbf{r}_i}{\partial q_j}(q_1, q_2, \cdots, q_n)\).↩︎