Action principle, Lagrangian mechanics, Hamiltonian mechanics, and conservation laws when assuming Aristotelian mechanics $F=mv$

Unlike Newtonian mechanics $$ m\ddot{q}^i~=~-\frac{\partial V(q)}{\partial q^i}, \tag{N}$$ the Aristotelian mechanics $$ m\dot{q}^i~=~-\frac{\partial V(q)}{\partial q^i} \tag{A}$$ is always dissipative and has no conventional stationary action principle. (See also this related Phys.SE post.) This implies that any attempt to define corresponding Aristotelian notions of Lagrangian & Hamiltonian mechanics, Noether current & conservation laws, are severely crippled from the onset.