This article studies a fully distributed optimal coordinated control problem with the global cost function for networked Euler-Lagrange (EL) systems subject to unknown model parameters. In particular, the global cost function is the sum of all the local cost functions assigned to each agent and only available to itself. The objective is to minimize the global cost function in a distributed manner while achieving a consensus on its optimal solution. Since the model parameters of the considered EL systems are not available, a new auxiliary system is introduced as a reference model, and its outputs exponentially converge the optimal solution of the global cost function. A fully distributed optimal control algorithm without requiring global information is first proposed. Then, an alternative distributed optimal algorithm via the event-triggered mechanism is proposed to reduce the communication cost. In particular, by combining an edge-based adaptive gain method, the proposed event-triggered optimal algorithm is also fully distributed. Finally, numerical simulation is carried out to validate the theoretical results.