A sufficient condition to solve an optimal control problem is to solve the Hamilton-Jacobi-Bellman (HJB) equation. However, finding a value function that satisfies the HJB equation for a nonlinear system is challenging. Inverse optimal control is an alternative method to solve the nonlinear optimal control problem by circumventing the need to solve the HJB equation. Inverse optimal adaptive control techniques have been developed that can handle structured (i.e., linear in the parameters (LP)) uncertainty for a particular class of nonlinear systems that do not include Euler-Lagrange systems with an uncertain time-varying inertia matrix. In this paper, an inverse optimal adaptive controller is developed to asymptotically minimize a meaningful performance index while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite LP uncertainty. A Lyapunov analysis is provided to examine the stability of the developed optimal controller, and preliminary simulation results illustrate the performance of the controller.