Consider a function $F=F(q)$ which is a symmetric, positive-definite matrix form of dimensions $n\times n$, where $q \in \mathbb{R}^n$ and $q=q(t)$.

When applying the midpoint integration rule on $F$, it is straightforward that $$ \int_0^h F(q)\,\mathrm{d}t \simeq h F\left(\frac{q_\ell + q_r}{2}\right) $$ where subindices $\ell$ and $r$ stand for "left" and "right" values respectively.

However, what would be the correct way to establish the following integral using the midpoint rule?

$$ \int_0^h \dot q^\intercal F(q)\, \dot q $$


I want to consider a centered finite difference for $\dot q$, such that

$$\dot q = \frac{d q}{dt} = \frac{q_r - q_\ell}{h},$$

where $h$ is the time step.

  • $\begingroup$ What is the differential equation for this second integral? $\ddot q=\dot q^TF(q)\dot q$? But then $F$ can't be a simple matrix. $\endgroup$ Commented Nov 17, 2023 at 16:18
  • $\begingroup$ @LutzLehmann Well, that is the point of the question. $F(q)$ is a square matrix whose elements are fuctions of $q$. $\endgroup$
    – Meclassic
    Commented Nov 18, 2023 at 17:05


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