I have the same confusion
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)
"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."