We formulate a Nash-based feedback control law for an Euler-Lagrange system to yield a solution to non-cooperative differential game. The robot manipulators are broadly utilized in industrial units on the account of their reliable, fast, and precise motions, while consuming a significant lumped amount of energy. Therefore, an optimal control strategy needs to be implemented in addressing efficiency issues, while delivering accuracy obligation. As a case study, we here focus on a 7-DOF robot manipulator through formulating a two-player feedback nonzero-sum differential game. First, coupled Euler-Lagrangian dynamic equations of the manipulator are briefly presented. Then, we formulate the feedback Nash equilibrium solution in order to achieve perfect trajectory tracking. Finally, the performance of the Nash-based feedback controller is analytically and experimentally examined. Simulation and experimental results reveal that the control law yields almost perfect tracking and achieves closed-loop stability.