I understand that Newton's Method is used to solve for the $(x_1, \ldots, x_n ) \in {\mathbf R}^n$ which ensures $f_1 = 0, \ldots, f_n = 0$ where $f_i$ is a nonlinear function of $(x_1, \ldots, x_n )$.
My problem is as follows:
Find the path $(x_1 (t), \ldots, x_n(t) ) \in {\mathbf R}^n$ for which $f_1 = 0, \ldots, f_n = 0$, where $f_i$ is a nonlinear function of $(x_1 (t), \ldots, x_n (t), t )$.
The question that I have is: Is it kosher to simply iterate through the independent variable $t$ and solve this system of nonlinear equations for fixed $t$ using Newton's Method, and then continue on, incrementing to the next value of $t$ that I want? This would imply that my system of equations is valid for infinitely many values of $t$, and that what happens at one $t$ does not effect the solution at another $t$, i.e. the $t$-sequenced systems of nonlinear equations are independent.
Thank you.
Your situation can be summarized as follows. Given a function $F : \mathbb{R}^{n+1} \rightarrow \mathbb{R}^n$, find a function $\gamma : \mathbb{R} \rightarrow \mathbb{R}^n$ such that $$F(t,\gamma(t)) = 0.$$ This is common problem and your approach is very natural. It is worth stressing the fact the solution $t \rightarrow \gamma(t)$ is very likely a continuous function of $t$. Therefore you can expect that the solution $x = \gamma(t)$ of $F(t,x) = 0$ is a very good initial guess for the Newton iteration that you will use to solve $F(t+\Delta t, x + \Delta x) = 0$. In short, you should process the chosen $t$ values in sequential order and use the solution from one system to initialize the search for the next solution.
In truth, I do not quite follow your reasoning during your last three lines, but I am not sure it is critical either. Perhaps my reply can help clarify your thoughts on this final point?