How do I use leapfrog method for a non-separable force ?
I know how to use it for the force of the form $f(q)+g(p)$. But I don’t know how to use it for the force like $f(q)\cdot g(p) $.
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Please explain your variables and the use of the word "force". – Lutz Lehmann Mar 11 '21 at 18:50
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@LutzLehmann it’s like q_dot = p and p_dot = func(q,p). I know what to do if func(q,p)= f(q) + g(p) but I don’t know what to do if func(q,p) = f(q)*g(p). (Also I want to solve the coupled system without combining two equations) – Sachin Vaidya Mar 11 '21 at 19:40
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So you do $q_{n+1}-q_n=hp_{n+1/2}$ and $p_{n+1/2}-p_{n-1/2}=hf(q_n)·g(0.5*(p_{n+1/2}+p_{n-1/2}))$ and recognize that the second is now an implicit equation. You could also look into the Numerov method, order 4 for a similar effort. – Lutz Lehmann Mar 11 '21 at 20:46
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@LutzLehmann actually the problem is functions f and g are mess and that’s why I wanted to avoid implicit method. – Sachin Vaidya Mar 12 '21 at 00:32