I recently started this question and it gave me some insight into the world of differential equations.
However the solution was not fit for my goals as I wanted a general method for calculating the position of a particle travelling through a vector field where the vectors in the field represents a force pushing on the particle.
So this is kind of a follow-up from the information I got from my other question.
So what do I have?
A vector field that is defined by a function:
$\vec{F}(x,y)$
I have left out the definition of the function since I wish to find a solution that works regardless off the function definition. But one example could be: $\vec{F}(x,y) = (\sin(x),\cos(y))$
A particle, which have some initial values for position and velocity
$\vec{p}(0)=P$
$\dot{\vec{p}}(0)=Q$
I know wish to find the position of the particle after a specified time $\vec{P}(t)$ and velocity $\dot{\vec{P}}(t)$.
From what I learned from the other question it's not possible to find a closed form solution to this, but you have to use some approximation with for example Euler's Method. However I don't know how to model this scenario as a differential equation and how to apply Euler's Method (or any other approximation technique) to my scenario. If someone could, step by step explain this process it would be much appreciated!
I should also add, the goal is to implement this in a program, so keep that in mind!
Thanks!
P.S.
I'm sure I missed stuff and you have questions about my logic here so I'd be happy to edit and answer any questions.