Jointly equivariant dynamics for interacting particles

Let a finite set of interacting particles be given, together with a symmetry Lie group G. Here we describe all
possible dynamics that are jointly equivariant with respect to the action of G. This is relevant e.g., when one
aims to describe collective dynamics that are independent of any coordinate change or external infuence.
We particularize the results to some key examples, i.e. for the most basic low dimensional symmetries
that appear in collective dynamics on manifolds.