I have a dynamical system whose state is a vector $\mathbf{y}\in \mathbf{R}^m$. The vectors $\mathbf{y},\dot{\mathbf{y}},\ddot{\mathbf{y}}\in \mathbf{R}^m$ relate to the vectors $\mathbf{z},\dot{\mathbf{z}},\ddot{\mathbf{z}}\in \mathbf{R}^n$ through the transformations
\begin{align}
\mathbf{y}&=g(\mathbf{z})\\
\dot{\mathbf{y}}&=J(\mathbf{z})\dot{\mathbf{z}}\\
\ddot{\mathbf{y}}&=J(\mathbf{z})\ddot{\mathbf{z}} + \dot{J}(\mathbf{z})\dot{\mathbf{z}}
\end{align}
where $J$ is the Jacobian matrix of the nonlinear transformation $g$.
What are the entries of $\dot{J}(\mathbf{z})$? For example, if $J_{ij}=\frac{\partial y_i}{\partial z_j}=a\sin(z_j)$, then what is $\dot{J}_{ij}$?
(There is a similar but unanswered question here.)
Best Answer
The Jacobian $J(\mathbf{z})$ is a matrix whose time derivative we are finding. The entries of $J(\mathbf{z})$ are expressed as \begin{equation} J_{ij}(\mathbf{z}) = \frac{\partial y_i}{\partial z_j}. \end{equation} I assume we can evaluate the entries of $J(\mathbf{z})$. For instance, in the example given in the question, we have that \begin{equation} J_{ij}(\mathbf{z})=a\sin(z_j) \end{equation} for some indices $i,j$. The time derivative of $J_{ij}(\mathbf{z})$ is \begin{align} \frac{\mathrm{d}}{\mathrm{d}t} J_{ij}(\mathbf{z}) &= \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial y_i}{\partial z_j}\\ &= \frac{\mathrm{d}}{\mathrm{d}t} a\sin(z_j)\\ &= a\dot{z}_j\cos(z_j). \end{align}