I am attempting to calculate the Jacobians for a system of differential equations which are being simulated via Verlet's method for solving differential equations numerically. The problem I am having is, how does one calculate the Jacobian for a map of the form:
$$\vec{f}(\vec{x_{n}},\vec{x_{n-1}})=\vec{x_{n+1}}$$
Ignoring the initial conditions, the Verlet Algorithm has the generic form:
$$\vec{x_{n+1}}=2\vec{x_{n}}-\vec{x_{n-1}}+\frac{\partial^{2}\vec{x_{n}}}{\partial t^{2}}h^{2}$$
For timestep h. Calculating the Jacobian involves taking a number of partial derivatives for each component of the map with respect to the coordinates that define $\vec{x}$. How does one deal with the derivatives of the terms of the form $x_{n-1}$? So if one of my coordinates was labelled $x$ and another was labelled $y$, what would the derivatives $\frac{\partial x_{n-1}}{\partial x}$ and $\frac{\partial x_{n-1}}{\partial y}$ look like?