I have a physical model: \begin{align} x''(t) &= a {\sqrt{x'(t)^2 + y'(t)^2}} x'(t) \\ y''(t) &= b {\sqrt{x'(t)^2 + y'(t)^2}} y'(t) + c \\ \end{align} or written differently (with speed as $v_x$ and $v_y$): \begin{align} \frac{\partial^2 x}{\partial t^2} = \frac{\partial v_x}{\partial t} &= a v v_x \\ \frac{\partial v_y}{\partial t} &= b v v_y + c\\ v &= {\sqrt{(v_x^2 + v_y^2)}} \end{align} I know a set of discrete points for $x(t)$ and $y(t)$ from $t = 1$ to $t = 2$.
This is the x and y curve over time from $t=0$ to $t=2$
QUESTION:
I need a numerical analysis to "go back in time" and approximate $x(t)$ and $y(t)$ from t=0 to t=1. How can I do that?
One idea is to calculate multiple Runge kutta simulations for different x(0) and y(0) values and find which one "fits best" the discrete values after t=1 but I was wondering if there would be a better method.
I'm interested by the maths but it's a numerical analysis so ultimately, I'm looking for pseudo code to solve this problem.
First, estimate the parameters $a,$ $b,$ and $c$ as well as you can.
Then set $u = -t$ and find the equations in terms of $u$ instead of $t.$
Then use your existing Runge-Kutta algorithm to integrate from $u = -1$ to $u = 0.$