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I was reading in a book that the angular momentum of the $N$-body system is of the form $Q(y,z)=y^TDz$ with $y, z$ vectors and $D$ a matrix. But the angular momentum of the $N$-body system is something like $L=\sum_{i=1}^Nq_i\times p_i$ no? How do you put it in the previous form?

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I am not sure if this helps but it is the citation that is causing me trouble.

We next consider partitioned Runge-Kutta methods for systems $\dot{y}=f(y,z), \dot{z}=g(y,z)$. Usually such methods cannot conserve general quadratic invariants. We therefore concentrate on quadratic invariants of the form \begin{equation} Q(y,z)=y^TDz \end{equation} where $D$ is a matrix of appropriate dimensions. Observe that the angular momentum of $N$-body systems is of this form.

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    $\begingroup$ You're going to have to specify a lot more information here because not everyone has access to the book you linked. You should just paraphrase the part that talks about this quantity. $\endgroup$
    – Triatticus
    Commented Dec 23, 2020 at 4:06
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    $\begingroup$ Given that angular momentum is a vector quantity, and a quadratic form like what you wrote is going to give you a scalar, there must be something more to explain about what is going on here. $\endgroup$
    – kaylimekay
    Commented Dec 23, 2020 at 9:16

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