In this paper, a very simple and efficient Adaptive Local Variational Iteration Method (LVIM) for solving problems of nonlinear dynamics and orbital mechanics is presented. The analytical iteration formula of this method is derived by using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian is required. Apart from the simple yet efficient iteration formula, a straightforward adaptive scheme is introduced to refine the step size and the collocation nodes at each time segment. The proposed adaptive method guarantees the prescribed accuracy without needing to manually tune the algorithm. It also prevents over-calculating in the computational process by relaxing the refinement automatically. Numerical results of large amplitude pendulum and the perturbed two-body problem validate the high accuracy and efficiency of this easy-to-use adaptive method.