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Wait.. where do we evaluate the velocity function?

I guess this would depend on whether you want to know if your approximation is upper or lower bounded by the true velocity? Or alternatively, we could also evaluate the velocity at the midpoint of the configurations, if they're linearly interpolatable. .


@lwan Yeah, it's definitely interesting to think about how this choice affects energy behavior. You're also totally right that you can evaluate it at the midpoint. Actually, even for nonlinear functions f something like this is possible. You can either use an update

(q_k+1 - q_k)/tau = f( (q_k + q_k+1)/2 )

and solve a nonlinear equation for q_k+1, or

(q_k+1 - q_k)/tau = ( f(q_k) + f(q_k+1) ) / 2

and solve a different nonlinear equation for q_k+1. The first rule is called the "midpoint rule" and the second one is called the "trapezoid." These will have different behavior (stability, accuracy, etc.) depending on exactly what system you're integrating. For instance, for certain systems the midpoint rule will exactly preserve the total energy of the system (offering an alternative to symplectic Euler).