In this paper, we show the equivalence between a constrained, multi-agent control problem, modeled within the port-Hamiltonian framework, and an exact potential game. Specifically, critical distance-based constraints determine a network of double-integrator agents, which can be represented as a graph. Virtual couplings, i.e., pairs of spring-damper, assigned to each edge of the graph, allow to synthesize a distributed, gradient-based control law that steers the network to an invariant set of stable configurations. We characterize the points belonging to such set as Nash equilibria of the associated potential game, relating the parameters of the virtual couplings with the equilibrium seeking problem, since they are crucial to shape the transient behaviour (i.e., the convergence) and, ideally, the set of reachable equilibria.
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