Timely paper on the pure mathematics of sloshing. This is a pre-print of a recent submission.

Dubois, Fran├žois, and Dimitri Stoliaroff. *"Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics."* arXiv preprint [arXiv:1407.1829](http://arxiv.org/pdf/1407.1829) (2014).

Imagine a volume with an equilibrium surface and then agitating it:


They consider a global vector, $q(t)$ for the sloshing representing elongation, rotation, and center of gravity and then elegantly describe it as this 2nd-order DiffEq (in the text as Eq. 84)

$ M \cdot \frac{d^2q}{dt^2} + K \cdot q =F(t) $

Looks familiar from the Mathieu DiffEq that I use above in #38, but is a vector representation and so $K$ is a matrix. If $K$ has a time dependence, then the quasi-periodic Mathieu/Hill dynamics emerge.