You might find this paper interesting. It does a similar decomposition with the dynamics of differentiable games (where the 'preferences' for how to change your strategy may not be the gradient of any function)
https://arxiv.org/abs/1802.05642
"The key result is to decompose the second-order dynamics into two components. The first is related to potential games, which reduce to gradient descent on an implicit function; the second relates to Hamiltonian games, a new class of games that obey a conservation law, akin to conservation laws in classical mechanical systems."
I have the same confusion